January 01, 1970
Under cone containment and the central ray hypothesis, we prove a determinant majorization result for hyperbolic polynomials on Euclidean Jordan algebras. In the case of real symmetric \(n \times n\) matrices, this recovers the main theorem of Harvey and Lawson [Duke Math. J. 174 (2025), no. 13, 2749–2763] and also yields a new \(\sigma_2\) majorization result. Moreover, for every \(3 \leq k \leq n-1\), we construct explicit hyperbolic polynomials satisfying cone containment and the central ray hypothesis but for which the analogous \(\sigma_k\) majorization fails.
Determinant majorization inequalities connect the theory of hyperbolic polynomials with fully nonlinear elliptic partial differential equations (PDEs). In matrix form, such an inequality compares a Gårding–Dirichlet operator \(F\) with the Monge-Ampère operator by an estimate of the form \[F(A)^{\frac{1}{N}} \geq F(I_n)^{\frac{1}{N}}(\det A)^{\frac{1}{n}}, \quad A>0,\] where \(N\) is the degree of \(F\). These inequalities have important applications in recent works on fully nonlinear PDEs; see, for example, [@Abja-Olive_AMPA2022; @Abja-Dinew-Olive_PA2023; @Guo-Phong_AOM2024_L_infinity_nonlinear; @Guo-Phong_CAG2024_Entropy_energy_bounds; @Guo-Phong-Tong_AOM2023_L_infinity_Monge-Ampere; @Harvey-Lawson_CVPDE2023_det_major].
Recall that a homogeneous polynomial \(p\) on a real vector space \(V\) is hyperbolic with respect to a direction \(a \in V\) if \(p(a)>0\) and, for every \(x \in V\), the one-variable polynomial \(t \mapsto p(ta + x)\) has only real roots. The corresponding Gårding cone \(\Gamma(p)_+\) is the connected component of \(\{p > 0\}\) containing \(a\). By Gårding’s theorem ([@Garding_JMM1959_inequality_hyperbolic]), \(p\) is hyperbolic with respect to every point of \(\Gamma(p)_+\). A Gårding–Dirichlet operator ([@Harvey-Lawson_CPAM2013_Garding]) is a hyperbolic polynomial on \(\undefined(n)\), the space of real symmetric \(n \times n\) matrices, such that its Gårding cone contains the cone of positive definite matrices.
In [@Harvey-Lawson_CVPDE2023_det_major], Harvey and Lawson proved determinant majorization for invariant Gårding-Dirichlet operators, including operators invariant under the real orthogonal, complex unitary, and quaternionic unitary groups, and also treated the Lagrangian Monge-Ampère operator. More recently in [@Harvey-Lawson_Duke2025_det_major], they proved that determinant majorization still holds when the invariance assumption is relaxed to the central ray hypothesis: the gradient \(\nabla F(I_n)\) is a positive multiple of \(I_n\). Operators satisfying this hypothesis are called \(I_n\)-central.
Theorem 1 ([@Harvey-Lawson_Duke2025_det_major]). For an \(I_n\)-central Gårding-Dirichlet operator \(F\) of degree \(N\) on \(\undefined(n)\), we have \[F(A)^{\frac{1}{N}} \geq F(I_n)^{\frac{1}{N}} (\det A)^{\frac{1}{n}} \quad \text{for every} \;\; A > 0.\]
The cone of positive definite real symmetric matrices is the basic example of a symmetric cone. Euclidean Jordan algebras provide an intrinsic framework for symmetric cones, with the Jordan determinant and Jordan trace playing the roles of the usual determinant and trace. This makes it natural to ask whether determinant majorization is a genuinely symmetric cone phenomenon rather than a special feature of matrix algebras. Our main result is the following.
Theorem 2. Let \(V\) be a finite-dimensional Euclidean Jordan algebra of rank \(r\) with unit element \(e\) and symmetric cone \(\undefined_+\). Suppose that \(p\) is a hyperbolic polynomial of degree \(m\) on \(V\) with Gårding cone \(\Gamma_+\). If \(\Gamma_+ \supset \undefined_+\) and \(p\) is \(e\)-central, that is, \(\tr^{p,e} = \frac{m}{r} \tr\nolimits_{\mathcal{J}}\), then \[p(a)^{\frac{1}{m}} \geq p(e)^{\frac{1}{m}} \left(\det\nolimits_{\mathcal{J}}a\right)^{\frac{1}{r}} \quad \text{for every} \;\; a \in \undefined_+.\]
When \(V = \undefined(n)\) with the standard Jordan product \(A\circ B=(AB+BA)/2\), the Jordan determinant is the usual determinant (see Example 2). Thus, in this case, Theorem 2 recovers Theorem 1.
The elementary symmetric function \(\sigma_2\) on \(\undefined^n\), defined by \[\sigma_2(x) = \sum_{i < j} x_i x_j, \quad x = (x_1, \dots, x_n) \in \undefined^n,\] is a hyperbolic polynomial whose Gårding cone is denoted by \(\Gamma(\sigma_2)_+\). The rank-two Lorentz Jordan algebra gives a symmetric cone realization of \(\Gamma(\sigma_2)_+\). By applying Theorem 2 to this model, we prove a \(\sigma_2\) majorization result in \(\undefined^n\). In the following, we denote \(\undefined= (1, \dots, 1) \in \undefined^n\).
Corollary 1. Suppose that \(p\) is a \(\undefined\)-central hyperbolic polynomial of degree \(m\) on \(\undefined^n\) with Gårding cone \(\Gamma(p)_+\). If \(\Gamma(p)_+ \supset \Gamma(\sigma_2)_+\), then \[p(a)^{\frac{1}{m}} \geq p(\undefined)^{\frac{1}{m}} \left(\frac{\sigma_2(a)}{{n \choose 2}}\right)^{\frac{1}{2}} \quad \text{for every} \;\; a \in \Gamma(\sigma_2)_+.\]
For \(A \in \undefined(n)\), we define \[\sigma_2(A) \mathrel{\vcenter{:}}= \sigma_2(\lambda_1(A), \dots, \lambda_n(A)),\] where \(\lambda_1(A), \dots, \lambda_n(A)\) are the usual eigenvalues of \(A\). Then \(\sigma_2\) is a hyperbolic polynomial on \(\undefined(n)\) and we denote its Gårding cone by \(\Gamma(\sigma_2)_+\). Analogously to Corollary 1, we also prove a \(\sigma_2\) majorization result in \(\undefined(n)\).
Corollary 2. Suppose that \(P\) is an \(I_n\)-central hyperbolic polynomial of degree \(m\) on \(\undefined(n)\) with Gårding cone \(\Gamma(P)_+\). If \(\Gamma(P)_+ \supset \Gamma(\sigma_2)_+\), then \[P(A)^{\frac{1}{m}} \geq P(I_n)^{\frac{1}{m}} \left(\frac{\sigma_2(A)}{{n \choose 2}}\right)^{\frac{1}{2}} \quad \text{for every} \;\; A \in \Gamma(\sigma_2)_+.\]
Although the determinant and \(\sigma_2\) majorization results hold for hyperbolic polynomials on \(\undefined^n\) and \(\undefined(n)\) satisfying the central ray hypothesis and cone containment, in Section 4, we show that this phenomenon for \(\sigma_2\) and \(\sigma_n\) does not extend to the intermediate elementary symmetric functions. Specifically, for all \(3 \leq k \leq n-1\), we construct explicit \(\undefined\)-central hyperbolic polynomials with the cone containment property but for which the analogous \(\sigma_k\) majorization statement fails.
In Section 5, we record equivalent gradient formulations of Theorem 2 and Corollaries 1 and 2 in terms of dual functions.
In this section, we recall some basic facts about Jordan algebras. Further background can be found in [@Faraut_book_Analysis_symmetric_cones].
A real vector space \(V\) is an algebra if there is a bilinear product \[V \times V \longmapsto V, \quad (x, y) \longmapsto x \circ y.\] An algebra \(V\) is called a Jordan algebra if it satisfies \[x \circ y = y \circ x \quad \text{and} \quad x \circ (x^2 \circ y) = x^2 \circ (x \circ y)\] for all \(x, y \in V\), where \(x^2 \mathrel{\vcenter{:}}= x \circ x\). When \(V\) is a Jordan algebra, we always assume that it has an element \(e\), called the unit element, which satisfies \[e \circ x = x \quad \text{for all} \;\; x \in V.\] We denote by \(\undefined[X]\) the algebra of one-variable real polynomials. The minimal polynomial of \(x \in V\) is the monic generator of the ideal \(\{p \in \undefined[X] : p(x) = 0\}\). Let \(\deg(x)\) denote the degree of the minimal polynomial of \(x\). The rank of \(V\) is defined by \[\rank(V) \mathrel{\vcenter{:}}= \max_{x \in V} \deg(x).\]
A real finite-dimensional Jordan algebra \(V\) is called Euclidean if there exists an inner product \(\langle \cdot, \cdot \rangle\) on \(V\) which satisfies the associative property: \[\langle x \circ y, z \rangle = \langle x, y \circ z \rangle \quad \text{for all} \;\; x, y, z \in V.\] A vector \(c \in V\) is called an idempotent if \[c^2 = c.\] Two idempotents \(c_1, c_2 \in V\) are called orthogonal if they satisfy \[c_1 \circ c_2 = 0.\] A nonzero idempotent is called primitive if it cannot be written as a sum of two nonzero orthogonal idempotents. A Jordan frame is a set \(\{c_1, \dots, c_k\}\) of pairwise-orthogonal primitive idempotents which satisfies \[c_1 + \dots + c_k = e.\]
An important fact for Euclidean Jordan algebras is the spectral theorem stated as follows.
Theorem 3 ([@Faraut_book_Analysis_symmetric_cones]). Let \(V\) be a finite-dimensional Euclidean Jordan algebra, and denote \(r = \rank(V)\). Then for every \(x \in V\), there exist a Jordan frame \(\{c_1, \dots, c_r\}\) and real numbers \(\lambda_1(x), \dots, \lambda_r(x)\) such that \[x = \sum_{i=1}^r \lambda_i(x) c_i.\] The numbers \(\lambda_i(x)\)’s, counted with multiplicities, are uniquely determined by \(x\).
For each \(x \in V\), the numbers \(\lambda_i(x)\)’s are called the Jordan eigenvalues of \(x\). We define \[\tr\nolimits_{\mathcal{J}} x \mathrel{\vcenter{:}}= \sum_{i=1}^r \lambda_i(x) \quad \text{and} \quad \det\nolimits_{\mathcal{J}} x \mathrel{\vcenter{:}}= \prod_{i=1}^r \lambda_i(x).\] By [@Faraut_book_Analysis_symmetric_cones], \(\tr\nolimits_{\mathcal{J}}\) and \(\det\nolimits_{\mathcal{J}}\) are polynomials on \(V\).
The cone of squares is the closed convex cone defined by \(\undefined\mathrel{\vcenter{:}}= \{x^2 : x \in V\}\). In spectral terms, we have \[\undefined= \{x \in V : \, \lambda_i(x) \geq 0 \;\; \text{for all} \;\; i = 1, \dots, r\}.\] The interior of \(\undefined\), called the symmetric cone of \(V\), is the open convex cone \[\undefined_+ \mathrel{\vcenter{:}}= \undefined(\undefined) = \{x \in V : \, \lambda_i(x) > 0 \;\; \text{for all} \;\; i = 1, \dots, r\}.\]
We are interested in the following models of Euclidean Jordan algebras.
Example 1 (\(\undefined^n\)). The Euclidean space \(\undefined^n\) is a Euclidean Jordan algebra with the Jordan product given by \[a \circ b = (a_1b_1, \dots, a_n b_n), \quad \text{for} \;\; a = (a_1, \dots, a_n) \;\; \text{and} \;\; b = (b_1, \dots, b_n).\] The unit element is \(\undefined\mathrel{\vcenter{:}}= (1, \dots, 1)\), and a Jordan frame is formed by the standard coordinate vectors. Thus for every \(x = (x_1, \dots, x_n) \in \undefined^n\), the Jordan eigenvalues of \(x\) are just \(x_1, \dots, x_n\). In particular, \[\tr\nolimits_{\mathcal{J}} x = x_1 + \dots + x_n \eqqcolon \sigma_1(x) \quad \text{and} \quad \det\nolimits_{\mathcal{J}} x = x_1 \dots x_n \eqqcolon \sigma_n(x).\] In addition, the symmetric cone is \(\undefined_+ = \undefined^n_+\).
Example 2 (\(\undefined(n)\)). Let \(\undefined(n)\) denote the space of \(n \times n\) real symmetric matrices. Then \(\undefined(n)\) is a Euclidean Jordan algebra equipped with the Jordan product given by \[A \circ B = \frac{AB + BA}{2} \quad \text{for} \;\; A, B \in \undefined(n).\] Clearly the unit element is \(I_n\), the identity matrix. In addition, a Jordan frame on \(\undefined(n)\) is \[c_i = u_i \otimes u_i, \quad i = 1, \dots, n,\] where \(u_1, \dots, u_n\) is an orthonormal basis of \(\undefined^n\). Thus, Jordan eigenvalues are just the usual matrix eigenvalues. In particular, \(\tr\nolimits_{\mathcal{J}}\) and \(\det\nolimits_{\mathcal{J}}\) coincide with the usual trace and determinant, respectively. The symmetric cone is \(\undefined_+ = \undefined(n)_+\), the cone of positive definite symmetric \(n \times n\) matrices.
The following special model of Euclidean Jordan algebras provides an important insight for the proof of Corollaries 1 and 2.
Example 3 (Lorentz). Let \(W\) be a finite-dimensional real vector space with an inner product \(\langle \cdot, \cdot \rangle_W\). The vector space \(V = \undefined\times W\) equipped with the product \[(t,u) \circ (s,v) = (ts + \langle u, v \rangle_W, tv + su)\] is a Euclidean Jordan algebra with unit element \(e = (1, 0)\). For every \(x = (t, u) \in V\), set \[\begin{cases*} c_1 = \frac{1}{2}\left(1, \frac{u}{|u|}\right), \; c_2 = \frac{1}{2}\left(1, -\frac{u}{|u|}\right) & if u \neq 0,\\ c_1 = \frac{1}{2}\left(1, \frac{v}{|v|}\right), \; c_2 = \frac{1}{2}\left(1, -\frac{v}{|v|}\right) & if u = 0, \end{cases*}\] where \(v\) is an arbitrary nonzero vector in \(W\). Then \((c_1, c_2)\) is a Jordan frame and \(x = (t,u) = (t+|u|)c_1 + (t-|u|)c_2\). Thus the Jordan eigenvalues of \(x = (t,u) \in V\) are \[\lambda_1(x) = t - |u| \quad \text{and} \quad \lambda_2(x) = t + |u|.\] Consequently, \(\rank(V) = 2\), and for \(x = (t, u) \in V\), \[\label{eq:32Lorentz32trace32and32det} \tr\nolimits_{\mathcal{J}} x = 2t \quad \text{and} \quad \det\nolimits_{\mathcal{J}} x = t^2 - |u|^2.\tag{1}\] The symmetric cone of \(V\) is \[\undefined_+ = \{(t,u) \in \undefined\times W : \, t > |u|\}.\]
We begin this section by providing some basic facts about hyperbolic polynomials. Further background can be found in [@Bauschke-Guler-Lewis-Sendov_CJM2001_Hyperbolic_poly; @Harvey-Lawson_CPAM2013_Garding; @Renegar_FoundCompMath2006_Hyperbolic_programs; @Wagner_BAMS2011_Multivariate_stable_polynomials]. We assume that \(V\) is a finite-dimensional real vector space. Let \(p\) be a homogeneous polynomial of degree \(m\) on \(V\). For any \(a, x \in V\) with \(p(a) \neq 0\), the one-variable polynomial \(s \mapsto p(sa + x)\) can be written as \[\label{eq:32chap32Garding32theory4432factor32of32homo32poly} p(sa + x) = p(a)\prod_{j=1}^m (s + \lambda^a_j(x)), \quad \lambda^a_j(x) \in \undefined\;\, \forall j.\tag{2}\] We define \(\lambda^a(x) \mathrel{\vcenter{:}}= (\lambda^a_1(x), \dots, \lambda^a_m(x)) \in \undefined^m\) modulo the permutation group \(\undefined_m\).
A homogeneous polynomial \(p\) of degree \(m\) on \(V\) is hyperbolic with respect to \(a\) (or \(a\)-hyperbolic), if \(p(a) > 0\) and \(\lambda^a(x) \in \undefined^m\) for every \(x \in V\). In this case, the Gårding cone \(\Gamma_+ \subset V\) (also denoted by \(\Gamma_+^a\), or \(\Gamma(p)_+\)) is defined by \[\Gamma_+ = \{x \in V :\, \lambda^a_j(x) > 0 \;\; \text{for all} \;\; j = 1, \dots, m\}.\] The closure of \(\Gamma_+\) in \(V\) is given by \[\Gamma = \overline{\Gamma_+} = \{x \in V : \lambda^a_j(x) \geq 0 \;\; \text{for all} \;\; j = 1, \dots, m\}.\] The fundamental result in the theory of hyperbolic polynomials is recorded in the following.
Theorem 4 ([@Garding_JMM1959_inequality_hyperbolic], see also [@Harvey-Lawson_CPAM2013_Garding; @Renegar_FoundCompMath2006_Hyperbolic_programs]). Suppose that \(p\) is \(a\)-hyperbolic and \(b \in \Gamma_+^a\). Then \(p\) is also \(b\)-hyperbolic and \(\Gamma_+^b = \Gamma_+^a\). Moreover, \(\Gamma^a_+\) is the connected component of \(\{p > 0\}\) containing \(a\), and is convex.
For an \(a\)-hyperbolic polynomial \(p\) of degree \(m\) on \(V\), we denote \[\tr^{p,a}(x) \mathrel{\vcenter{:}}= \sum_{j=1}^{m} \lambda^{p,a}_j(x) \quad \text{for} \;\; x \in V.\]
Definition 1 (Central ray hypothesis). Let \(V\) be a finite-dimensional Euclidean Jordan algebra with the unit element \(e\). An \(e\)-hyperbolic polynomial \(p\) of degree \(m\) on \(V\) is said to be \(e\)-central if there is a constant \(\kappa > 0\) such that \[\tr^{p,e}(x) = \kappa \tr\nolimits_{\mathcal{J}}(x) \quad \text{for all} \;\; x \in V.\] By homogeneity, the constant \(\kappa\) is necessarily \(\kappa = \frac{m}{\rank(V)}\).
Further useful properties of the central ray hypothesis for hyperbolic polynomials in an arbitrary vector space can be obtained by analogously following the arguments in [@Harvey-Lawson_CVPDE2023_det_major] (where vector space is \(\undefined(n)\)).
Example 4. Let \(V\) be a finite-dimensional Euclidean Jordan algebra of rank \(r\) with unit element \(e\) and symmetric cone \(\undefined_+\). The determinant function \(\det\nolimits_{\mathcal{J}}\) on \(V\) is an \(e\)-central hyperbolic polynomial of degree \(r\) whose Gårding cone is \(\Gamma_+ = \undefined_+\). Indeed, for each \(x \in V\), let \(c_1, \dots, c_r\) be a Jordan frame such that \[x = \sum_{i=1}^r \lambda_i(x) c_i.\] Then for every \(t \in \undefined\), we have \[\det\nolimits_{\mathcal{J}}(te + x) = \det\nolimits_{\mathcal{J}}\left(\sum_{i=1}^r (t + \lambda_i(x))c_i\right) = \prod_{i=1}^r (t+\lambda_i(x)).\] Thus \(\det\nolimits_{\mathcal{J}}\) is \(e\)-hyperbolic and \(\Gamma_+ = \undefined_+\). The \(e\)-central condition is obvious.
A hyperbolic polynomial in \(\undefined^n\) is said to be stable if its Gårding cone contains \(\undefined^n_+\). Stable hyperbolic polynomials have only nonnegative coefficients (see [@Choe-Oxley-Sokal-Wagner_AdvApplMath2004_Homogeneous_multivariate_polynomials; @Wagner_BAMS2011_Multivariate_stable_polynomials]). The combination of this fact and [@Harvey-Lawson_CVPDE2023_det_major] (see also [@Gurvits_EJC2008_hyperbolic_poly]) gives the following result which plays a crucial role in the proof of the main result.
Lemma 1. Suppose that \(p\) is a \(\undefined\)-central stable hyperbolic polynomial of degree \(m\) on \(\undefined^n\). Then for every \(x = (x_1, \dots, x_n) \in \undefined^n_+\), we have \[p(x)^{\frac{1}{m}} \geq p(\undefined)^{\frac{1}{m}} (x_1 \dots x_n)^{\frac{1}{n}}.\]
Note that Lemma 1 is the same as Theorem 2 when \(V = \undefined^n\).
In this subsection, we prove the main result and its consequences. The proof of Theorem 2 follows the same strategy as the proof of [@Harvey-Lawson_Duke2025_det_major].
Proof of Theorem 2. We begin by fixing \(a \in \undefined_+ \subset \Gamma_+\). By Theorem 3, there exists a Jordan frame \(c_1, \dots, c_r \in V\) such that \[a = \sum_{i=1}^r \alpha_i c_i,\] where \(\alpha_1, \dots, \alpha_r > 0\) are the Jordan eigenvalues of \(a\). We consider a real polynomial in \(\undefined^r\) defined by \[Q(x) \mathrel{\vcenter{:}}= p(x_1 c_1 + \dots + x_r c_r) \quad \text{for} \;\; x = (x_1, \dots, x_r) \in \undefined^r.\] For convenience, we define the linear map \(L : \undefined^r \to V\) by \[L(x) = \sum_{i=1}^r x_i c_i \quad \text{for} \;\; x = (x_1, \dots, x_r) \in \undefined^r.\] In particular, \(L(\undefined) = e\), the unit element of \(V\), and \(Q(x) = p(L(x))\). Important properties of \(Q\) are proved in the following claims.
Claim 1: \(Q\) is \(m\)-homogeneous and \(Q(\undefined) = p(e)\).
Proof of Claim: Since \(p\) is \(m\)-homogeneous, for any \(s > 0\) and \(x \in \undefined^r\), we have \[Q(sx) = p(L(sx)) = p(sL(x)) = s^m p(L(x)) = s^m Q(x).\] Moreover, \(Q(\undefined) = p(L(\undefined)) = p(e)\).
Claim 2: \(Q\) is stable.
Proof of Claim: Let \(v = (v_1, \dots, v_r) \in \undefined^r_+\). Then since \((c_1, \dots, c_r)\) is a Jordan frame, the point \(b =L(v)\) has
eigenvalues \(v_1, \dots, v_r > 0\), so \(b \in \undefined_+\). Since \(\undefined_+ \subset \Gamma_+\) by the assumption, \(p\) is \(b\)-hyperbolic. Consequently, \(Q(v) = p(b) > 0\) and for every \(x \in \undefined^r\), the one-variable polynomial
\[t \mapsto Q(tv + x) = p(tb + L(x))\] has only real roots. Therefore \(Q\) is \(v\)-hyperbolic. Since \(v \in
\undefined^r_+\) is arbitrary and \(\undefined^r_+\) is connected, by Theorem 4, we conclude that \(Q\) is stable.
Claim 3: \(Q\) is \(\undefined\)-central.
Proof of Claim: Note that for every \(x \in \undefined^r\), the proof of Claim 2 yields \[\lambda^{Q,\undefined}(x) = \lambda^{p,e}(L(x)).\] Combining this and the assumption that
\(p\) is \(e\)-central, for every \(x \in \undefined^r\), we have \[\begin{align}
\tr^{Q, \undefined}(x) = \tr^{p,e}(L(x)) &= \frac{m}{r} \tr\nolimits_{\mathcal{J}}(L(x)) \\
&= \frac{m}{r}(x_1 + \dots + x_r).
\end{align}\] This proves that \(Q\) is \(\undefined\)-central.
By these properties of \(Q\), we apply Lemma 1 to conclude that for every \(x = (x_1, \dots, x_r) \in \undefined^r_+\), \[p(L(x))^\frac{1}{m} = Q(x)^{\frac{1}{m}} \geq Q(\undefined)^{\frac{1}{m}}(x_1 \dots x_r)^{\frac{1}{r}} = p(e)^{\frac{1}{m}}(x_1 \dots x_r)^{\frac{1}{r}}.\] In particular, at \(x = (\alpha_1, \dots, \alpha_r) \in \undefined^r_+\), we get \[p(a)^{\frac{1}{m}} \geq p(e)^{\frac{1}{m}}(\alpha_1 \dots \alpha_r)^{\frac{1}{r}} = p(e)^{\frac{1}{m}} \left(\det\nolimits_{\mathcal{J}}a\right)^{\frac{1}{r}}.\] The proof is complete. ◻
We now prove Corollaries 1 and 2. The idea is to apply Theorem 2 to the Lorentz model of Euclidean Jordan algebras (Example 3).
Proof of Corollary 1. We denote \(W = \undefined^\perp = \{y \in \undefined^n : \langle y, \undefined\rangle = 0\}\). By the orthogonal decomposition \(\undefined^n = \undefined\undefined\oplus W\), for every \(x \in \undefined^n\), we write \[x = t\undefined+ y \quad \text{where} \;\; t =\frac{\sigma_1(x)}{n} \;\; \text{and} \;\; y \in W.\] Then for every \(x \in \undefined^n\), we have \(|x|^2 = nt^2 + |y|^2\) and \[\begin{align} \sigma_2(x) = \frac{\sigma_1(x)^2}{2} - \frac{|x|^2}{2} &= \frac{\sigma_1(x)^2}{2} - \frac{nt^2 + |y|^2}{2}\\ &= \frac{n(n-1)}{2} t^2 - \frac{|y|^2}{2}. \end{align}\] Thus \[\label{eq:32sigma95232and32det32as32Jordan32algebra} \frac{\sigma_2(x)}{{n \choose 2}} = t^2 - \frac{|y|^2}{n(n-1)} = t^2 - |u|^2,\tag{3}\] where we denote \(u = \frac{y}{\sqrt{n(n-1)}} \in W\). Consequently, \[\begin{align} \Gamma(\sigma_2)_+ &\equiv \{x : \, \sigma_1(x) > 0, \; \sigma_2(x) > 0\}\\ &= \{(t,u) : \, t > 0, \; t^2 - |u|^2 > 0\}\\ &= \{(t,u) : \, t > |u|\}. \end{align}\] Thus \(\Gamma(\sigma_2)_+\) can be identified with the symmetric cone \(\undefined_+\) of the Lorentz Jordan algebra \(V = \undefined\undefined\oplus W\) (see Example 3) under the identification map \(x \mapsto (t, u)\). In this way, the unit element \(e = (1,0)\) of \(V\) is identified with the vector \(\undefined\in \undefined^n\).
Now, let \(p\) be a \(\undefined\)-central hyperbolic polynomial of degree \(m\) on \(\undefined^n\) whose Gårding cone contains \(\Gamma(\sigma_2)_+\). Then by the identification map \(x \mapsto (t, u)\), \(p\) is also an \(e\)-hyperbolic polynomial on \(V\) whose Gårding cone (in \(V\)) contains \(\undefined_+\). Moreover, since \(p\) is \(\undefined\)-central in \(\undefined^n\), for every \(x \in \undefined^n\), we have \[\tr^{p, e}(t, u) = \tr^{p, \undefined}(x) = \frac{m}{n} \sigma_1(x) = mt = \frac{m}{2} \tr\nolimits_{\mathcal{J}}(t,u),\] where we have used 1 to obtain the last equality. Thus \(p\) is also \(e\)-central in \(V\). Applying Theorem 2 for \(p\) in \(V\), we get \[p(t,u)^{\frac{1}{m}} \geq p(e)^{\frac{1}{m}} \left(\det\nolimits_{\mathcal{J}}(t,u)\right)^{\frac{1}{2}} \quad \text{for every} \;\; (t,u) \in \undefined_+.\] Recalling 3 and the fact that \(\det\nolimits_{\mathcal{J}}(t,u) = t^2 - |u|^2\) by 1 , we conclude that \[p(x)^{\frac{1}{m}} \geq p(\undefined)^{\frac{1}{m}} \left(\frac{\sigma_2(x)}{{n \choose 2}}\right)^{\frac{1}{2}} \quad \text{for every} \;\; x \in \Gamma(\sigma_2)_+.\] The proof is complete. ◻
Proof of Corollary 2. The proof is similar to that of Corollary 1. We begin by denoting \(W = \{Y \in \undefined(n) : \sigma_1(Y) = 0\}\). By the orthogonal decomposition \(\undefined(n) = \undefined I_n \oplus W\), for every \(X \in \undefined(n)\), we write \[X = tI_n + Y \quad \text{where} \;\; t =\frac{\sigma_1(X)}{n} \;\; \text{and} \;\; Y \in W.\] Then for every \(X \in \undefined(n)\), we have \(|X|^2 = nt^2 + |Y|^2\) and \[\begin{align} \sigma_2(X) = \frac{\sigma_1(X)^2}{2} - \frac{|X|^2}{2} &= \frac{\sigma_1(X)^2}{2} - \frac{nt^2 + |Y|^2}{2}\\ &= \frac{n(n-1)}{2} t^2 - \frac{|Y|^2}{2}. \end{align}\] Thus \[\label{eq:32sigma95232and32det32as32Jordan32algebra32matrices} \frac{\sigma_2(X)}{{n \choose 2}} = t^2 - \frac{|Y|^2}{n(n-1)} = t^2 - |U|^2,\tag{4}\] where we denote \(U = \frac{1}{\sqrt{n(n-1)}}Y \in W\). Consequently, \[\begin{align} \Gamma(\sigma_2)_+ &\equiv \{X : \, \sigma_1(X) > 0, \; \sigma_2(X) > 0\}\\ &= \{(t,U) : \, t > 0, \; t^2 - |U|^2 > 0\}\\ &= \{(t,U) : \, t > |U|\}. \end{align}\] Thus \(\Gamma(\sigma_2)_+\) can be identified with the symmetric cone \(\undefined_+\) of the Lorentz Jordan algebra \(V = \undefined I_n \oplus W\) (see Example 3) under the identification map \(X \mapsto (t, U)\). In this way, the unit element \(e = (1,0)\) of \(V\) is identified with the identity matrix \(I_n\).
Now, let \(P\) be a \(I_n\)-central hyperbolic polynomial of degree \(m\) on \(\undefined(n)\) whose Gårding cone contains \(\Gamma(\sigma_2)_+\). Then by the identification map \(X \mapsto (t, U)\), \(P\) is also an \(e\)-hyperbolic polynomial on \(V\) whose Gårding cone (in \(V\)) contains \(\undefined_+\). Moreover, since \(P\) is \(I_n\)-central in \(\undefined(n)\), for every \(X \in \undefined(n)\), we have \[\tr^{P, e}(t, U) = \tr^{P, I_n}(X) = \frac{m}{n} \sigma_1(X) = mt = \frac{m}{2} \tr\nolimits_{\mathcal{J}}(t,U),\] where we have used 1 to obtain the last equality. Thus \(P\) is also \(e\)-central in \(V\). Applying Theorem 2 for \(P\) in \(V\), we get \[P(t,U)^{\frac{1}{m}} \geq P(e)^{\frac{1}{m}} \left(\det\nolimits_{\mathcal{J}}(t,U)\right)^{\frac{1}{2}} \quad \text{for every} \;\; (t,U) \in \undefined_+.\] Recalling 4 and the fact that \(\det\nolimits_{\mathcal{J}}(t,U) = t^2 - |U|^2\) by 1 , we conclude that \[P(X)^{\frac{1}{m}} \geq P(I_n)^{\frac{1}{m}} \left(\frac{\sigma_2(X)}{{n \choose 2}}\right)^{\frac{1}{2}} \quad \text{for every} \;\; X \in \Gamma(\sigma_2)_+.\] The proof is complete. ◻
For each \(1 \leq k \leq n\), the elementary symmetric function \(\sigma_k\) on \(\undefined^n\), defined by \[\sigma_k(x) = \sum_{i_1 < \dots < i_k} x_{i_1} \dots x_{i_k} \quad \text{for} \;\; x = (x_1, \dots, x_n) \in \undefined^n,\] is a \(\undefined\)-central hyperbolic polynomial. We denote its Gårding cone by \(\Gamma(n,k)_+\).
For each \(1 \leq k \leq n\) and \(X \in \undefined(n)\), we define \[\sigma_k(X) \mathrel{\vcenter{:}}= \sigma_k(\lambda(X)),\] where \(\lambda(X) \mathrel{\vcenter{:}}= (\lambda_1(X), \dots, \lambda_n(X))\) (defined modulo the permutation group \(\mathfrak{S}_n\)) is the vector whose coordinates are the usual matrix eigenvalues of \(X\). Then \(\sigma_k\) is an \(I_n\)-central hyperbolic polynomial on \(\undefined(n)\). We denote its Gårding cone by \(\Gamma_{\undefined}(n,k)_+\). Then \[\Gamma_{\undefined}(n,k)_+ = \{X \in \undefined(n) : \lambda(X) \in \Gamma(n,k)_+\}.\]
The main point of this section is that for \(3 \leq k \leq n-1\), the central ray hypothesis together with cone containment is not sufficient for hyperbolic \(\sigma_k\) majorization, either on \(\undefined^n\) or on \(\undefined(n)\), in contrast with Corollary 1 and Lemma 1. After recalling two useful facts about elementary symmetric functions, we construct \(\undefined\)-central (\(I_n\)-central) hyperbolic polynomials that violate \(\sigma_k\) majorization for every \(3 \leq k \leq n-1\).
Lemma 2 (see e.g. [@Lin-Trudinger_BAMS1994_Inequalities_elem_sym_funct]). Let \(2 \leq k \leq n\). If \(x = (x_1, \dots, x_n) \in \Gamma(n,k)_+\), then \[(x_1, \dots, x_{n-\ell}) \in \Gamma(n-\ell, k-\ell)_+ \quad \text{for all} \;\; 1 \leq \ell \leq k-1.\] In particular, if \(x = (x_1, \dots, x_n) \in \Gamma(n,k)_+\), then \[x_1 + \dots + x_{n-k+1} > 0.\]
Before stating the next lemma, we define, for \(1 \leq k \leq n\), the linear function \(L_k\) on \(\undefined^n\) by \[L_k(x) = \frac{x_1 + \dots + x_{n-k+1}}{n-k+1} \quad \text{for} \;\; x \in \undefined^n.\]
Lemma 3. For each \(2 \leq k \leq n\), we have \[\frac{\sigma_1(x)}{n} \geq \frac{n-k+1}{2n-k} L_k(x) \quad \text{for all} \;\; x \in \Gamma(n,k)_+.\]
Proof. Let \(x = (x_1, \dots, x_n) \in \Gamma(n,k)_+\). By [@Kuo-Trudinger_IUMJ2007_New_max_principle], we have \[x_i \geq - \frac{n-k}{n(k-1)} \sigma_1(x) \quad \text{for each} \;\; i = 1, \dots, n.\] Summing this over the last \((k-1)\) coordinates yields \[\sum_{i=n-k+2}^{n} x_i \geq - \frac{n-k}{n} \sigma_1(x).\] Therefore, \[(n-k+1)L_k(x) = \sigma_1(x) - \sum_{i=n-k+2}^{n} x_i \leq \frac{2n-k}{n} \sigma_1(x).\] ◻
We now construct \(\undefined\)-central (\(I_n\)-central) hyperbolic polynomials violating \(\sigma_k\) majorization for all \(n \geq 4\) and \(3 \leq k \leq n-1\).
Example 5 (\(\undefined^n\)). For \(2 \leq k \leq n\), note that \[\frac{n-k+1}{2n-k} > \frac{1}{k} \quad \Longleftrightarrow \quad 2 < k < n.\] Let \(3 \leq k \leq n-1\). We fix a rational number \(q = \frac{r}{m}\), \(r, m \in \mathbb{N}\), such that \[\frac{1}{k} < q < \frac{n-k+1}{2n-k}.\] Define the linear function \(L\) on \(\undefined^n\) by \[L(x) = \frac{1}{1-q} \left(\frac{\sigma_1(x)}{n} - q L_k(x)\right) \quad \text{for} \;\; x \in \undefined^n.\] Then \(L(x) > 0\) for all \(x \in \Gamma(n,k)_+\) by Lemma 3, and \(L(\undefined) = 1\). We consider the polynomial \(p\) on \(\undefined^n\) given by \[p(x) = L_k(x)^r L(x)^{m-r} \quad \text{for} \;\; x \in \undefined^n.\] Then \(p\) is \(m\)-homogeneous, \(p(\undefined) = 1\), and for all \(t \in \undefined\), \(x \in \undefined^n\), \[p(t\undefined+ x) = (t + L_k(x))^r (t + L(x))^{m-r}.\] Thus \(p\) is \(\undefined\)-hyperbolic with Gårding cone \[\Gamma(p)_+ = \{x \in \undefined^n : L_k(x) > 0, \; L(x) > 0\}.\] Consequently, \(\Gamma(p)_+ \supset \Gamma(n,k)_+\) since \(L(x) > 0\) and \(L_k(x) > 0\) (by Lemma 2) for all \(x \in \Gamma(n,k)_+\). Moreover, \(p\) is \(\undefined\)-central since \[\tr^{p, \undefined}(x) = r L_k(x) + (m-r) L(x) = m \left(qL_k(x) + (1-q)L(x) \right) = \frac{m \sigma_1(x)}{n}.\]
We put \(\alpha = n-k+1\), \(\beta = k-1\), and consider the point \[x_{\varepsilon} = (\underbrace{\varepsilon, \dots, \varepsilon}_{\alpha}, \underbrace{1, \dots, 1}_{\beta}), \quad \varepsilon > 0.\] Then \(x_{\varepsilon} \in \undefined^n_+ \subset \Gamma(n,k)_+\). Moreover, \(L_k(x_\varepsilon) = \varepsilon\), and \[L(x_{\varepsilon}) = \frac{\frac{\alpha\varepsilon + \beta}{n} - q\varepsilon}{1-q} \to \frac{\beta}{n(1-q)} \quad \text{as} \;\; \varepsilon \to 0.\] Consequently, \(p(x_{\varepsilon})^{\frac{1}{m}} \sim C \varepsilon^q.\) Meanwhile, \(\sigma_k(x_{\varepsilon}) = \alpha \varepsilon + O(\varepsilon^2)\), so \(\sigma_k(x_{\varepsilon})^{\frac{1}{k}} \sim C'\varepsilon^{\frac{1}{k}}\). Since \(q > \frac{1}{k}\), this implies that for sufficiently small \(\varepsilon > 0\), we have \[p(x_{\varepsilon})^{\frac{1}{m}} < p(\undefined)^{\frac{1}{m}} \left(\frac{\sigma_k(x_\varepsilon)}{{n \choose k}}\right)^{\frac{1}{k}}.\] So \(\sigma_k\) majorization fails for \(p\).
Example 6 (\(\undefined(n)\)). Let \(3 \leq k \leq n-1\) and recall the hyperbolic polynomial \(p\) of degree \(m\) on \(\undefined^n\) constructed in Example 5. For each \(X = (X_{ij}) \in \undefined(n)\), we denote \[d(X) \mathrel{\vcenter{:}}= (X_{11}, \dots, X_{nn}).\] Consider the polynomial \(F\) on \(\undefined(n)\) defined by \[F(X) = p(d(X)) \quad \text{for} \;\; X \in \undefined(n).\] Then for every \(t \in \undefined\) and \(X \in \undefined(n)\), we have \[F(tI_n + X) = p(t\undefined+ d(X)).\] Thus \(F\) is \(I_n\)-hyperbolic with \[\lambda^{F, I_n}(X) = \lambda^{p, \undefined}(d(X)).\] It follows that \[\tr^{F, I_n}(X) = \tr^{p, \undefined}(d(X)) = \frac{m\sigma_1(d(X))}{n} = \frac{m \tr(X)}{n},\] which means that \(F\) is \(I_n\)-central. In addition, \[\label{eq:32example32of32matrix32nonmajor4432Garding32cone} \Gamma(F)_+ = \{X \in \undefined(n) : d(X) \in \Gamma(p)_+\}.\tag{5}\] To see cone containment, let \(X \in \Gamma_{\undefined}(n,k)_+\). Then \(\lambda(X) \in \Gamma(n,k)_+\). Moreover, by Schur-Horn theorem and Birkhoff’s theorem (see [@Horn-Johnson_Book_Matrix]), we have \[d(X) \in \mathrm{conv}\{\pi(\lambda(X)) : \pi \in \mathfrak{S}_n\}.\] Since \(\Gamma(n,k)_+\) is convex and invariant under coordinate permutations, we conclude that \(d(X) \in \Gamma(n,k)_+ \subset \Gamma(p)_+\). By 5 , this implies \(X \in \Gamma(F)_+\). Hence \(\Gamma(F)_+ \supset \Gamma_{\undefined}(n,k)_+\).
Now, we consider the positive definite diagonal matrix \[X_{\varepsilon} \mathrel{\vcenter{:}}= \mathrm{diag}(\underbrace{\varepsilon, \dots, \varepsilon}_{n-k+1}, \underbrace{1, \dots, 1}_{k-1}), \quad \varepsilon > 0.\] Following the argument in Example 5, for sufficiently small \(\varepsilon > 0\), we have \[F(X_{\varepsilon})^{\frac{1}{m}} < F(I_n)^{\frac{1}{m}} \left(\frac{\sigma_k(X_\varepsilon)}{{n \choose k}}\right)^{\frac{1}{k}}.\] So \(\sigma_k\) majorization fails for \(F\).
We note, however, that the polynomial constructed in Example 5 (resp. Example 6) is not invariant under the permutation group \(\mathfrak{S}_n\) (resp. the real orthogonal group \(\mathrm{O}(n)\)). Thus, if one additionally assumes the invariance condition, either on \(\undefined^n\) or on \(\undefined(n)\), it remains inconclusive whether the analogous \(\sigma_k\) majorization holds for \(3 \leq k \leq n-1\).
In this section, we always assume that \(p\) is a hyperbolic polynomial of degree \(m\) on a finite-dimensional real vector space \(V\). The Gårding cone of \(p\) is denoted by \(\Gamma_+\) and its closure in \(V\) is denoted by \(\Gamma\). Define \[f(x) = p(x)^{\frac{1}{m}}, \quad x \in \Gamma.\] Then \(f\) is \(1\)-homogeneous on \(\Gamma\). Moreover, \(f\) and \(\log f\) are concave in \(\Gamma_+\). Specifically, by [@Harvey-Lawson_CVPDE2023_det_major], we have \[[D^2 f(a)](v,v) = -\frac{f(a)}{m^2} \Big(m|\lambda^a(v)|^2 - \sigma_1(\lambda^a(v))^2\Big) \quad \text{for all} \;\; a \in \Gamma_+ \;\; \text{and} \;\; v \in V,\] and \[\label{eq:32Hessian32of32log32hyperbolic} [D^2 \log f(a)](v, v) = - \frac{|\lambda^a(v)|^2}{m} \quad \text{for all} \;\; a \in \Gamma_+ \;\; \text{and} \;\; v \in V.\tag{6}\]
The edge \(\undefined\) of the Gårding cone \(\Gamma_+\) (or \(\Gamma\)) is defined by \[\undefined\mathrel{\vcenter{:}}= \Gamma \cap (-\Gamma).\] The edge \(\undefined\) is also the set of all \(x \in V\) such that \(\lambda^a(x) = 0\) for some \(a \in \Gamma_+\), and hence for all \(a \in \Gamma_+\) (see e.g. [@Harvey-Lawson_CPAM2013_Garding]).
The Gårding cone \(\Gamma_+\) (or \(\Gamma\)) is said to be regular1 if its edge \(\undefined= \{0\}\). If \(\Gamma\) is regular, then the dual cone of \(\Gamma\), \[\Gamma^* \mathrel{\vcenter{:}}= \{u \in V^* :\, \langle u, x \rangle \geq 0 \;\; \text{for all} \;\; x \in \Gamma\},\] is full-dimensional, and whose interior, \(\Gamma^*_+\), satisfies \(\Gamma^* = \overline{\Gamma^*_+}\). Moreover, the bipolar theorem gives \((\Gamma^*)^* = \Gamma\). Here, we identify \((V^*)^*\) with \(V\).
Proposition 5. Suppose that \(p\) is a hyperbolic polynomial on \(V\) with Gårding cone \(\Gamma_+\). If \(\Gamma\) is regular, then \(\nabla \log f : \Gamma_+ \to \Gamma^*_+\) is a real-analytic diffeomorphism.
Proof. Assume \(\Gamma\) is regular. By [@Renegar_FoundCompMath2006_Hyperbolic_programs] and its proof, \(\nabla \log f : \Gamma_+ \to \Gamma^*_+\) is a bijection. In addition, \(\nabla \log f\) is real-analytic in \(\Gamma_+\) since \(p\) is a polynomial which is positive in \(\Gamma_+\). Thus, by 6 and the inverse function theorem, \(\nabla \log f : \Gamma_+ \to \Gamma^*_+\) is a real-analytic diffeomorphism. ◻
We recall the following elementary inequality which is of great importance for the theory of hyperbolic polynomials. A proof (taken from [@Harvey-Lawson_CPAM2013_Garding]) is provided here for the sake of exposition.
Lemma 4. For any \(a, x \in \Gamma_+\), we have \[\frac{f(x)}{f(a)} \leq \langle \nabla \log f(a), x \rangle,\] with equality if and only if \(\lambda^a_1(x) = \dots = \lambda^a_m(x)\).
Proof. Let \(a \in \Gamma_+\). We recall 2 that \[\label{eq:32chap32Garding32theory4432factor32of32homo32poly32again} p(sa + x) = p(a)\prod_{j=1}^m (s + \lambda^a_j(x)) \quad \text{for every} \;\; s \in \undefined\;\; \text{and} \;\; x \in V.\tag{7}\] In particular, taking \(s = 0\) in 7 , we have \[\label{eq:32p40x41326132p40a4132product32eigenvalues} p(x) = p(a) \prod_{j=1}^m \lambda^a_j(x) \quad \text{for every} \;\; x \in V.\tag{8}\] Furthermore, since \(p\) is \(m\)-homogeneous, one can see from 7 that \[\lambda^a(tx) = t\lambda^a(x) \quad \text{for every} \;\; t \in \undefined\;\; \text{and} \;\; x \in V.\] Thus, taking \(s = 1\) and replacing \(x\) by \(tx\) in 7 , we get \[\label{eq:32factor32p40a3243tx41} p(a + tx) = p(a) \prod_{j=1}^m (1 + t \lambda^a_j(x)) \quad \text{for all} \;\, t \in \undefined\text{ and } x \in V.\tag{9}\] Then, taking the logarithmic derivative of 9 at \(t=0\) yields \[\label{eq:32chap32Garding32theory4432trace32as32gradient32of32log} \langle \nabla \log p(a), x\rangle = \sum_{j=1}^{m} \lambda^a_j(x) \quad \text{for every} \;\; x \in V.\tag{10}\] If \(x \in \Gamma_+\), then \(\lambda^a_j(x) > 0\) for each \(j = 1, \dots, m\). Therefore, since \(f(x) = p(x)^{\frac{1}{m}}\), the assertion follows from 8 , 10 and the AM-GM inequality. ◻
The dual function \(f^*\) of \(f\) is defined by \[\label{eq:32dual32function} f^*(u) \mathrel{\vcenter{:}}= \inf_{x \in \Gamma_+} \frac{\langle u, x \rangle}{f(x)} = \inf_{x \in \Gamma_+, \, f(x) = 1} \langle u,x \rangle, \quad \text{for} \;\; u \in \Gamma^*.\tag{11}\] In the context of hyperbolic polynomials, \(f^*\) has several useful properties listed in the following.
Theorem 6. Assume that the Gårding cone \(\Gamma_+\) is regular. The following properties of \(f^*\) hold.
If \(u \in \Gamma^*_+\), then \(f^*(u) > 0\) and the infimum in 11 is attained: \[f^*(u) = \min_{x \in \Gamma_+, \, f(x) = 1} \langle u, x \rangle = \frac{1}{f(a)} = \frac{1}{f((\nabla \log f)^{-1}(u))},\] where \(a \in \Gamma_+\) satisfies \(\nabla \log f(a) = u\), and the minimum is attained only at \(x = \frac{a}{f(a)}\).
\(f^*\) is real-analytic in \(\Gamma^*_+\).
\(f^*(\nabla f(a)) = 1\) for every \(a \in \Gamma_+\).
If \(u \in \partial \Gamma^* \backslash \{0\}\), then \(f^*(u) = 0\), so the infimum in 11 is not attained at any point in \(\Gamma_+\).
\(f^*\) is concave, continuous, and \(1\)-homogeneous on \(\Gamma^*\).
The basic inequality is \[\langle u, a \rangle \geq f^*(u) f(a) \quad \text{for all} \;\; u \in \Gamma^*_+ \;\; \text{and} \;\; a \in \Gamma_+,\] with equality if and only if \(\nabla f(a) = \frac{u}{f^* (u)}\) or, equivalently, \(\nabla f^*(u) = \frac{a}{f(a)}\).
The gradients are related by \[\nabla f^*(\nabla f(a)) = \frac{a}{f(a)}, \quad \nabla f(\nabla f^*(u)) = \frac{u}{f^*(u)} \quad \; \text{for all} \;\; u \in \Gamma^*_+ \;\; \text{and} \;\; a \in \Gamma_+.\] In particular, \((f \nabla f) = (f^* \nabla f^*)^{-1}\) in \(\Gamma_+\).
\((f^*)^* = f\).
Proof. We prove these statements one by one.
Proof of (a): Let \(u \in \Gamma^*_+\). Since \(\Gamma\) is regular, by Proposition 5, there exists a unique point \(a \in \Gamma_+\) such that \(\nabla
\log f(a) = u\). By Lemma 4, it follows that \[\frac{\langle u, x
\rangle}{f(x)} = \frac{\langle \nabla \log f(a), x \rangle}{f(x)} \geq \frac{1}{f(a)} \quad \text{for every} \;\; x \in \Gamma_+,\] with equality if and only if \(\lambda_1^a(x) = \dots = \lambda_m^a(x)\). Clearly,
\(x = a/f(a)\) satisfies the equality condition. In addition, suppose \(y \in \Gamma_+\) satisfies \(\lambda^a(y) = c\undefined\) for some \(c > 0\). Then by the fact that \[\lambda^a(ta + x) = t\undefined+ \lambda^a(x) \quad \text{for all} \;\; t \in \undefined\;\; \text{and} \;\; x \in V,\] which can be obtained from 7 , we deduce that \(\lambda^a(y - ca) = 0\). This means \(y-ca \in \undefined\). Since \(\Gamma\) is regular, we must have \(y = ca\).
Proof of (b): This follows from (a) and Proposition 5.
Proof of (c): Note that since \(f\) is concave and \(1\)-homogeneous, we have \[\frac{\langle \nabla f(a), x \rangle}{f(x)} \geq 1 \quad \text{for every}
\;\; x \in \Gamma_+,\] where the equality holds at \(x = a\).
Proof of (d): Let \(u \in \partial \Gamma^* \backslash \{0\}\). Then \(\langle u, x \rangle \geq 0\) for all \(x \in \Gamma\), so \(f^*(u) \geq 0\). In addition, since \(u \in \partial \Gamma^*\), by an interior cone condition (see e.g. [@Faraut_book_Analysis_symmetric_cones]), there exists \(z \in \Gamma\backslash\{0\}\) such that \(\langle u, z \rangle = 0\). If
\(z \in \Gamma_+\), then there exists \(\varepsilon > 0\) such that \(z + \varepsilon u \in \Gamma_+\) and \(z - \varepsilon u
\in \Gamma_+\). It follows that \[\varepsilon |u|^2 = \langle u, z + \varepsilon u \rangle \geq 0 \quad \text{and} \quad -\varepsilon |u|^2 = \langle u, z - \varepsilon u \rangle \geq 0.\] This contradiction implies
that \(z \in \partial \Gamma\), so \(p(z) = 0\). Let \(a \in \Gamma_+\) and let \(r\) be the rank of \(z\), i.e., \[r = \#\{j \in [m] : \lambda_j^a(z) \neq 0\}.\] Then \(r \geq 1\) since \(\Gamma\) is regular and \(z \neq 0\). Thus by 9 , we have \[p(a + tz) = p(a)\prod_{j=1}^m (1 + t \lambda^a_j(z)) \geq Ct^r \quad \text{for all} \;\; t \geq 0,\] for some
constant \(C > 0\). Since \(a + tz \in \Gamma_+\) for all \(t \geq 0\), it follows that \[f^*(u) \leq \frac{\langle u, a + tz
\rangle}{f(a + tz)} = \frac{\langle u,a \rangle}{f(a + tz)} \to 0 \quad \text{as} \;\; t \to \infty.\] Hence \(f^*(u) = 0\), and the infimum in 11 is not attained at any point in
\(\Gamma_+\).
Proof of (e): As the pointwise infimum of a family of linear functions, \(f^*\) is \(1\)-homogeneous and concave in \(\Gamma^*\). In addition, it is
clear that \(f^*\) is continuous at \(0\). It remains to prove the continuity of \(f^*\) at \(\partial \Gamma^* \backslash
\{0\}\).
Let \(a \in \Gamma_+\) and \(u \in \partial \Gamma^* \backslash \{0\}\). The argument in the proof of (d) shows that there exists \(z \in \partial\Gamma \backslash \{0\}\) satisfying \(\langle u, z \rangle = 0\). Let \(r\) be the rank of \(z\). Then since \(p(z) = 0\) and \(\Gamma\) is regular, we have \(1 \leq r \leq m-1\). Moreover, by 9 , we have \[p(a + tz) = p(a)\prod_{j=1}^m (1 + t \lambda^a_j(z)) \geq Ct^r \quad \text{for all} \;\; t \geq 0,\] for some constant \(C > 0\). Hence, \[\label{eq:32estimate32f40a43tz4132for32z32on32boundary} f(a+tz) = p(a+tz)^{\frac{1}{m}} \geq Ct^{\frac{r}{m}} \quad \text{for all} \;\; t \geq 0,\tag{12}\] for some constant \(C > 0\).
Suppose that \(\{u_k\}_{k \in \mathbb{N}}\) is a sequence in \(\Gamma^*\) such that \(u_k \to u\) as \(k \to \infty\).
Since \(\require{physics}
\eval{f^*}_{\partial\Gamma^*} = 0\) by (d), we need to prove \(f^*(u_k) \to 0\). It suffices to assume \(u_k \in \Gamma^*_+\) for all \(k \in
\mathbb{N}\). Then setting \(\delta_k = \langle u_k, z \rangle\), by an interior cone condition (see e.g. [@Faraut_book_Analysis_symmetric_cones]), we have \(\delta_k > 0\) and \[\delta_k \to \langle u, z \rangle = 0 \quad \text{as} \;\; k
\to \infty.\] Furthermore, by 12 , for every \(t \geq 0\), since \(a + tz \in \Gamma_+\), we get
\[\label{eq:32decay32of32f40uk41}
f^*(u_k) \leq \frac{\langle u_k, a + tz \rangle}{f(a + tz)} = \frac{\langle u_k, a \rangle + t\delta_k}{f(a + tz)} \leq Ct^{-\frac{r}{m}} + C \delta_k t^{1-\frac{r}{m}},\tag{13}\] for some constant \(C >
0\). Choosing any \(0 < \alpha < \frac{m}{m-r}\) and then setting \(t = \delta_k^{-\alpha}\) in 13 , we obtain \[f^*(u_k) \leq C \delta_k^{\frac{\alpha r}{m}} + C \delta_k^{1 - \alpha(1 - \frac{r}{m})} \to 0 \quad \text{as} \;\; k \to \infty.\] This completes the proof of (e).
Proof of (f): The inequality of (f) simply follows from the definition 11 . Moreover, if the equality holds, i.e., \(\langle u, a \rangle = f^*(u) f(a)\) for some \(u \in \Gamma^*_+\) and \(a \in \Gamma_+\), then \[f(a) = \left\langle \frac{u}{f^*(u)}, a \right\rangle \quad \text{while} \quad f(x) \leq \left\langle
\frac{u}{f^*(u)}, x \right\rangle \;\; \text{for all} \;\; x \in \Gamma_+.\] Since \(f\) is concave and \(1\)-homogeneous, this implies that \(\nabla f(a) =
u/f^*(u)\). By (b) and (e), this argument also shows that \(\nabla f^*(u) = a/f(a).\)
Proof of (g): Note that for any \(a \in \Gamma_+\) and \(u \in \Gamma^*_+\), by (f) we have \[\nabla f(a) = \frac{u}{f^*(u)} \quad \Longleftrightarrow
\quad \nabla f^*(u) = \frac{a}{f(a)}.\] We combine this and the fact that \(\nabla f\) and \(\nabla f^*\) are \(0\)-homogeneous to conclude (g).
Proof of (h): Let \(x \in \Gamma_+\). The basic inequality in (f) gives \[f(x) \leq \frac{\langle u, x \rangle}{f^*(u)} \quad \text{for every} \;\; u \in \Gamma^*_+.\] Taking the
infimum over \(u \in \Gamma^*_+\) of the right-hand side yields \(f(x) \leq (f^*)^*(x)\). To see the reverse inequality, take \(u = \nabla f(x)\). Then \(u \in \Gamma^*_+\) by Proposition 5 and \[(f^*)^*(x) \leq \frac{\langle u, x \rangle}{f^*(u)} = \frac{\langle \nabla f(x), x \rangle}{f^*(\nabla f(x))} =
f(x),\] where we have used (c) to obtain the last equality. Therefore \((f^*)^*(x) = f(x)\).
For \(x \in \partial\Gamma\), we let \(a \in \Gamma_+\). For any \(\varepsilon > 0\), since \(x + \varepsilon a \in \Gamma_+\), we have \[(f^*)^*(x + \varepsilon a) = f(x + \varepsilon a).\] Since \(\langle a, u \rangle \geq 0\) for all \(u \in \Gamma^*_+\), this implies \[0 \leq (f^*)^*(x) = \inf_{u \in \Gamma^*_+} \frac{\langle u, x \rangle}{f^*(u)} \leq \inf_{u \in \Gamma^*_+} \frac{\langle u, x + \varepsilon a \rangle}{f^*(u)} = (f^*)^*(x + \varepsilon a) = f(x + \varepsilon a).\] Since \(f(x) = 0\), we take the limit as \(\varepsilon \to 0\) in the above inequalities to conclude that \((f^*)^*(x) = 0 = f(x)\).
The proof is complete. ◻
Example 7. Let \(V\) be a finite-dimensional Euclidean Jordan algebra of rank \(r\) with unit element \(e\), symmetric cone \(\undefined_+\), and the inner product \[\langle x, y \rangle = \tr\nolimits_{\mathcal{J}}(x \circ y), \quad x, y \in V.\] Example 4 shows that \(\det\nolimits_{\mathcal{J}}\) is hyperbolic in \(V\) with Gårding cone \(\Gamma_+ = \undefined_+\). Moreover, under the identification \(V^* = V\) via the inner product, the symmetric cone is self-dual, i.e., \(\Gamma^*_+ = \Gamma_+ = \undefined_+\). Define the function \[f(x) = \left(\det\nolimits_{\mathcal{J}}x\right)^{\frac{1}{r}} \quad \text{for} \;\; x \in \undefined.\] For \(a \in \Gamma_+ = \undefined_+\), [@Faraut_book_Analysis_symmetric_cones] give \(\nabla \log \det\nolimits_{\mathcal{J}}(a) = a^{-1}\), so \[\nabla \log f(a) = \frac{1}{r} \nabla \log \det\nolimits_{\mathcal{J}}(a) = \frac{a^{-1}}{r}.\] In other words, for every \(u \in \Gamma^*_+ =\undefined_+\), setting \(a = (ru)^{-1}\) gives \(\nabla \log f(a) = u\). Thus, by Theorem 6(a), we have \[f^*(u) = \frac{1}{f(a)} = \frac{1}{f((ru)^{-1})} = \frac{1}{\det\nolimits_{\mathcal{J}}((ru)^{-1})^\frac{1}{r}} = \left(\det\nolimits_{\mathcal{J}}(ru)\right)^{\frac{1}{r}} = r \left(\det\nolimits_{\mathcal{J}}u\right)^{\frac{1}{r}}.\]
Example 8. This example is taken from [@Kuo-Trudinger_IUMJ2007_New_max_principle]. We consider the function \(\sigma_2\) on \(\undefined^n\). The dual cone of \(\Gamma(\sigma_2)\) is \[\Gamma^* = \left\{u \in \undefined^n : \, \sigma_1(u) \geq \sqrt{(n-1)}|u| \right\}.\] Put \(f(x) = \sigma_2(x)^{\frac{1}{2}}\) for \(x \in \Gamma(\sigma_2)\). Then for every \(u \in \Gamma^*\), we have \[f^*(u) = \sqrt{2} \left(\frac{\sigma_1(u)^2}{n-1} - |u|^2\right)^{\frac{1}{2}}.\]
Example 9. This example serves as an analogue of Example 8 for symmetric matrices. We consider the function \(\sigma_2\) on \(\undefined(n)\). The dual cone of \(\Gamma(\sigma_2)\) is \[\Gamma^* = \left\{A \in \undefined(n) : \, \tr(A) \geq \sqrt{n-1}|A| \right\}.\] Put \(f(X) = \sigma_2(X)^{\frac{1}{2}}\) for \(X \in \Gamma(\sigma_2)\). Then for every \(A \in \Gamma^*\), we have \[f^*(A) = \sqrt{2} \left(\frac{\tr(A)^2}{n-1} - |A|^2\right)^{\frac{1}{2}}.\]
Remark 7. Let \(p\) and \(q\) be hyperbolic polynomials in \(V\) of degree \(m_p\) and \(m_q\), respectively, and suppose that \(\Gamma(p)_+ \supset \Gamma(q)_+ \supset \{e\}\). We define \[f(x) = p(x)^{\frac{1}{m_p}} \;\; \text{for } x \in \Gamma(p) \quad \text{and} \quad g(y) = q(y)^{\frac{1}{m_q}} \;\; \text{for } y \in \Gamma(q).\] Since \(f\) is \(1\)-homogeneous and concave in \(\Gamma(p)_+\), we have \[f(a) \leq \langle \nabla f(x), a \rangle \quad \text{for all} \;\; a,x \in \Gamma(p)_+,\] where the equality holds at \(x = a\). Therefore, the following equivalences hold \[\begin{align} \frac{f(a)}{f(e)} \geq \frac{g(a)}{g(e)} \;\; \forall\, a \in \Gamma(q)_+ & \;\; \Longleftrightarrow \;\; \frac{\langle \nabla f(x), a \rangle}{g(a)} \geq \frac{f(e)}{g(e)} \;\; \forall\, x \in \Gamma(p)_+,\, \forall\, a \in \Gamma(q)_+\\ & \;\; \Longleftrightarrow \;\; \inf_{a \in \Gamma(q)_+} \frac{\langle \nabla f(x), a \rangle}{g(a)} \geq \frac{f(e)}{g(e)} \;\; \forall\, x \in \Gamma(p)_+\\ & \;\; \Longleftrightarrow \;\; g^*(\nabla f(x)) \geq \frac{f(e)}{g(e)} \;\; \forall\, x \in \Gamma(p)_+. \end{align}\]
Remark 7 and Example 7 show that Theorem 2 equivalently gives the following generalization of [@Harvey-Lawson_Duke2025_det_major].
Corollary 3. Let \(V\) be a finite-dimensional Euclidean Jordan algebra of rank \(r\) with unit element \(e\) and symmetric cone \(\undefined_+\). Suppose that \(p\) is an \(e\)-central hyperbolic polynomial of degree \(m\) on \(V\) with Gårding cone \(\Gamma_+\). We set \(f(x) = p(x)^{\frac{1}{m}}\) for \(x \in \Gamma_+\). If \(\Gamma_+ \supset \undefined_+\), then \[\det\nolimits_{\mathcal{J}}(\nabla f(x)) \geq \left(\frac{f(e)}{r}\right)^r \quad \text{for all} \;\; x \in \Gamma_+.\]
Similarly, by Remark 7, Example 8, and Corollary 1, we have the following.
Corollary 4. Let \(p\) be a \(\undefined\)-central hyperbolic polynomial of degree \(m\) on \(\undefined^n\) with Gårding cone \(\Gamma(p)_+\). We set \(f(x) = p(x)^{\frac{1}{m}}\) for \(x \in \Gamma(p)_+\). If \(\Gamma(p)_+ \supset \Gamma(\sigma_2)_+\), then \[\sigma_1(\nabla f(x))^2 - (n-1)|\nabla f(x)|^2 \geq \frac{f(\undefined)^2}{n} \quad \text{for all} \;\; x \in \Gamma(p)_+.\]
In the symmetric matrix setting, Remark 7, Example 9, and Corollary 2 give the following analogue.
Corollary 5. Let \(P\) be an \(I_n\)-central hyperbolic polynomial of degree \(m\) on \(\undefined(n)\) with Gårding cone \(\Gamma(P)_+\). We set \(F(X) = P(X)^{\frac{1}{m}}\) for \(X \in \Gamma(P)_+\). If \(\Gamma(P)_+ \supset \Gamma(\sigma_2)_+\), then \[\tr(\nabla F(X))^2 - (n-1)|\nabla F(X)|^2 \geq \frac{F(I_n)^2}{n} \quad \text{for all} \;\; X \in \Gamma(P)_+.\]
The author was supported by National Key R&D Program of China 2020YFA0712800.
This is called complete in [@Harvey-Lawson_CPAM2013_Garding].↩︎