January 01, 1970
Myroshnychenko, Tatarko, and Yaskin constructed a body \(K\) in \(\mathbb{R}^n\), \(n \geq 5\), with the property that there is exactly one hyperplane \(H\) passing through \(c(K)\), the centroid of \(K\), such that the centroid of \(K\cap H\) coincides with \(c(K)\). This construction provided answers to questions of Grünbaum and Loewner for \(n\geq 5\), which are still open in dimensions \(3\) and \(4\). We study analogues of these questions in the settings of hyperbolic space \(\mathbb{H}^n\) and \(s\)-concave functions on \(\mathbb{R}^n\).
Let \(K\) be a convex body in \(\mathbb{R}^n\), i.e., a compact convex set with non-empty interior. The centroid of \(K\) is the point \[\label{def-centroid} c(K) = \frac{1}{|K|} \int_K x \, dx,\tag{1}\] where integration is with respect to Lebesgue measure and \(|K|\) denotes the volume of \(K\).
Consider the family of hyperplanes in \(\mathbb{R}^n\) containing the centroid of \(K\). If \(K\) is centrally symmetric then \(c(K\cap H)=c(K)\) for each hyperplane \(H\) from this family. For general convex bodies this is no longer true. It is natural to ask how many sections with the property \(c(K\cap H)=c(K)\) every convex body \(K\) in \(\mathbb{R}^n\) has. The following problems were posed by Grünbaum [1] and Loewner [2]; see also [3].
Problem 1. (Grünbaum) Is the centroid \(c(K)\) of \(K \subset \mathbb{R}^n\) the centroid of at least \(n + 1\) different \((n-1)\)-dimensional sections of \(K\) through \(c(K)\)?
Problem 2. (Loewner) Let \(\mu(K)\) be the number of hyperplane sections of \(K\) passing through \(c(K)\) whose centroid is the same as \(c(K)\). Let \(\mu(n) =\displaystyle \min_{K\in \mathcal{K}^n} \mu(K)\) where \(\mathcal{K}^n\) is the class of all convex bodies in \(\mathbb{R}^n\). What is the value of \(\mu(n)\)?
It is easy to show that \(\mu(2)=3\); as was noticed by Grünbaum [1] and Loewner [2]. If \(n\ge 3\), Grünbaum [1] has shown that \(\mu(n)\ge 1\); see also [4]. Myroshnychenko, Tatarko, and Yaskin [5] have shown that \(\mu(n)=1\) for \(n\ge 5\). The case of dimensions \(n=3\) and \(n=4\) is still open. It is natural to study the problem in other settings, where we can obtain the answer in all dimensions. In this paper we show that, in hyperbolic space \(\mathbb{H}^n\) and in the case of \(s\)-concave functions on \(\mathbb{R}^n\) with \(-1/(n+1)<s<\infty\), the analogue of the number \(\mu(n)\) equals 1 for all \(n\ge 3\).
For other recent results about centroids of convex bodies the reader is referred to [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].
We will start with the hyperboloid model of hyperbolic space \(\mathbb{H}^n\). The reader is referred to [19] for additional background information. Let \(\mathbb{R}^{n,1}\) be the Minkowski space, which can be identified with \(\mathbb{R}^{n+1}\) equipped with the Minkowski inner product \[\langle x, y\rangle_{n,1} =-x_0y_0+x_1y_1+\cdots+x_ny_n,\] where \(x=(x_0,x_1, \ldots, x_n)\) and \(y=(y_0,y_1, \ldots, y_n)\).
In the hyperboloid model, we define \[\mathcal{H}^n=\{x\in\mathbb{R}^{n,1}:\;\langle x, x\rangle_{n,1}=-1,\;x_0>0\},\] endowed with the induced metric \(g_{\mathcal{H}}\). The corresponding volume element we denote by \(d\mathrm{vol}_{\mathcal{H}}\).
For a set \(L\subset\mathcal{H}^n\) of positive volume, define its moment vector \[\label{eq:Z-def} Z(L)=\int_{L} x\, d\mathrm{vol}_{\mathcal{H}}(x) = (Z_0(L),\boldsymbol{Z}(L)) \in\mathbb{R}^{n,1}\tag{2}\] where \(Z_0(L) \in \mathbb{R}\) and \(\boldsymbol{Z}(L) \in \mathbb{R}^n\). This point is not on \(\mathcal{H}^n\), but if we normalize it properly, then we will get the center of mass of \(L\). Let \[\label{eq:m-def} m(L)=\sqrt{-\langle Z(L),Z(L)\rangle_{n,1}} =\sqrt{Z_0(L)^2-|\boldsymbol{Z}(L)|^2}.\tag{3}\] Here and throughout the paper, for a vector \(p \in \mathbb{R}^n\), \(|p|\) denotes its Euclidean norm. Then the centroid of \(L\) is the point in \(\mathcal{H}^n\) defined by \[\label{eq:C-on-H} C(L)=\frac{Z(L)}{m(L)}.\tag{4}\] This definition of centroid in \(\mathcal{H}^n\) is analogous to the one in the spherical space (see [20] for a study of centroids for discrete sets in constant curvature spaces).
Instead of the hyperboloid model, it will be more convenient to use the Poincaré model of \(\mathbb{H}^n\) in the unit ball \(\mathbb{B}^n\) in \(\mathbb{R}^n\). The identification between the two models can be described geometrically as follows. Consider the unit Euclidean ball \(\mathbb{B}^n\) in the hyperplane \(\{x_0=0\} \subset \mathbb{R}^{n,1}\). Given a point \(x\) in \(\mathcal{H}^n\), consider the line segment connecting \(x\) and \((-1,0,\ldots, 0)\), The line segment intersects \(\mathbb{B}^n\) at some point \(p\). The corresponding map \(F:\mathbb{B}^n\to\mathcal{H}^n\), that sends \(p\) to \(x\), is given by \[\label{eq:Cayley} F(p)=\Big(\frac{1+|p|^2}{1-|p|^2},\,\frac{2p}{1-|p|^2}\Big), \qquad p\in\mathbb{B}^n.\tag{5}\]
Using \(F\) to pull back the metric from \(\mathcal{H}^n\), we obtain the metric in the Poincaré model: \[\label{eq:Poincare-metric} g_{\mathbb{B}}=\frac{4}{(1-|p|^2)^2}\sum_{i=1}^n dp_i^2,\tag{6}\] and hence its volume element is \[\label{eq:volB} d\mathrm{vol}_{\mathbb{B}} = \frac{2^n}{(1-|p|^2)^n}\,dp,\tag{7}\] where \(dp\) denotes Lebesgue measure on \(\mathbb{R}^n\).
Any two points in the Poincaré model can be connected by a unique geodesic segment. A set \(K\subset \mathbb{B}^n\) is called convex if for any \(p_1\) and \(p_2\) in \(K\), the geodesic segment connecting these points lies in \(K\). Let \(K\) be a convex body in the Poincaré model, i.e., a compact convex set with non-empty interior. We define the centroid of \(K\) as the preimage of 4 under the map \(F\).
From 2 , using 5 and 7 , we obtain \[\begin{align} Z_0(K) &= \int_{K}\frac{2^n\,(1+|p|^2)}{(1-|p|^2)^{n+1}}\,dp,\tag{8}\\ \boldsymbol{Z}(K) &= \int_{K}\frac{2^{n+1}\,p}{(1-|p|^2)^{n+1}}\,dp.\tag{9} \end{align}\] From 5 we see that if \(x=F(p)\), then \(p_i=x_i/(x_0+1)\), \(i=1,\ldots, n\). Thus
\[\label{eq:ball-centroid} C(K)=\frac{\boldsymbol{Z}(K)/m(K)}{Z_0(K)/m(K)+1}=\frac{\boldsymbol{Z}(K) }{Z_0(K)+m(K)},\tag{10}\] where \(m(K)=\sqrt{Z_0(K)^2-|\boldsymbol{Z}(K)|^2}\).
For any vector \(\xi\) on the unit sphere \(S^{n-1}\), consider the hypersurface \(\xi_{\mathbb{B}}^\perp\) in \(\mathbb{B}^n\) passing through the origin defined by \[\xi_{\mathbb{B}}^\perp = \{x\in \mathbb{B}^n: \xi_1 x_1+ \cdots+\xi_n x_n=0\}.\] Such a hypersurface is a totally geodesic submanifold in the Poincaré ball model of hyperbolic space (they are analogous to hyperplanes in the Euclidean space in the sense that every geodesic line on such a surface is also a geodesic in the ambient space).
Let \(K\) be a convex body in \(\mathbb{B}^n\) that contains the origin in its interior. For any \(\xi\in S^{n-1}\), the section of \(K\) by the hypersurface \(\xi_{\mathbb{B}}^\perp\) is a convex body in \(\xi_{\mathbb{B}}^\perp\) and its centroid can be expressed analogously to 10 , with \(n\) replaced by \(n-1\): \[\label{eq:section-centroid} C(K\cap \xi_{\mathbb{B}}^\perp)= \frac{1}{Z_0(K\cap\xi_{\mathbb{B}}^\perp)+m(K\cap \xi_{\mathbb{B}}^\perp)} \int_{K\cap \xi_{\mathbb{B}}^\perp}\frac{2^{n}\,p}{(1-|p|^2)^{n}}\,dp,\tag{11}\]
Analogous to Problem 2, we can consider the following question in hyperbolic space.
Problem 3. Let \(K\) be a convex body in \(\mathbb{H}^n\) and \(\eta(K)\) be the number of totally geodesic \((n-1)\)-dimensional sections of \(K\) passing through \(c(K)\) whose centroid is the same as \(c(K)\). Let \(\eta(n) =\displaystyle \min_{K\in \mathcal{K}({\mathbb{H}}^n)} \eta(K)\) where \(\mathcal{K}({\mathbb{H}}^n)\) is the class of all convex bodies in \(\mathbb{H}^n\). What is the value of \(\eta(n)\)?
Since the body \(K\subset \mathbb{B}^n\) can be identified with a body in \(\mathbb{R}^n\), we can apply to \(K\) standard Euclidean concepts. We say that a compact set \(K \subset \mathbb{R}^n\) is star-shaped about the origin \(0\) if for every point \(x \in K\) each point of the interval \([0, x)\) is an interior point of \(K\). The Minkowski functional of \(K\) is defined by \[\|x\|_K = \min\{\lambda \geq 0: x \in \lambda K \}.\] We say that \(K\) is a star body if it is compact, star-shaped about the origin and its Minkowski functional is a continuous function on \(\mathbb{R}^n\).
The radial function of a star body \(K\) is defined by \[\rho_K(\xi) = \max \{\lambda > 0: \lambda \xi \in K \}, \quad \xi \in S^{n-1}.\] Observe that \(\rho_K(\xi) = \|\xi\|_K^{-1}\) for any \(\xi \in S^{n-1}\) and \(\rho_K\) is positive and continuous on \(S^{n-1}\).
We say that \(K\) is origin symmetric if \(x\in K \Leftrightarrow -x \in K\). For an origin symmetric star body \(K\), its radial function \(\rho_K\) is an even function on the sphere, i.e., \(\rho_K(\xi)=\rho_K(-\xi)\) for all \(\xi\in S^{n-1}\).
Let us now discuss functional versions of Problem 2. Let \(-\infty \le s\le \infty\). A function \(f: \mathbb{R}^n\to \mathbb{[}0,\infty)\) is called \(s\)-concave if \[\label{s-concave} f(\lambda x + (1-\lambda)y)\ge \left(\lambda f^s(x)+ (1-\lambda)f^s(y)\right)^{1/s},\tag{12}\] for all \(x, y\in \mathbb{R}^n\) such that \(f(x)\cdot f(y)>0\) and all \(\lambda\in (0,1)\).
If \(s =-\infty\), \(0\), \(\infty\), the definition above is understood in the sense of limits. In particular, \(f\) is \(\infty\)-concave if \[f(\lambda x + (1-\lambda)y)\ge\max\{f(x),f(y)\},\] for all \(x, y\in \mathbb{R}^n\) such that \(f(x)\cdot f(y)>0\) and all \(\lambda\in (0,1)\). Such functions are constant multiples of indicator functions of convex sets.
If \(s=0\), inequality 12 becomes \[f(\lambda x + (1-\lambda)y)\ge f^\lambda(x) f^{1-\lambda}(y),\] for all \(x, y\in \mathbb{R}^n\) and all \(\lambda\in (0,1)\). Such functions are called log-concave.
We will denote by \(C_s(\mathbb{R}^n)\) the class of \(s\)-concave functions on \(\mathbb{R}^n\) with positive finite integrals. It is known that these classes become larger when \(s\) gets smaller.
Below we will focus on functions from \(C_s(\mathbb{R}^n)\) with \(-1/(n+1)<s<\infty\). If \(f\in C_s(\mathbb{R}^n)\) with \(s\ge 0\), then \[f(x)\le Ae^{-B|x|},\] for all \(x\in \mathbb{R}^n\) and some positive constants \(A\) and \(B\); see [21]. If \(-1/(n+1)<s<0\), then there is a constant \(C>0\), such that \[\label{s-conc} f(x)\le \frac{C}{1+|x|^{-1/s}},\tag{13}\] for all \(x\in \mathbb{R}^n\); see [22]. Thus, if \(f\in C_s(\mathbb{R}^n)\) with \(-1/(n+1)<s<\infty\), then its first moments exist, and we can define its centroid (or barycenter) analogously to 1 : \[c(f) = \frac{\int_{\mathbb{R}^n} x f(x)\, dx}{\int_{\mathbb{R}^n} f(x)\, dx} .\] If \(H\) is an affine subspace of \(\mathbb{R}^n\), then we denote by \[c_H(f)=\frac{\int_{H} x f(x)\, dx}{\int_{ H} f(x)\, dx}\] the centroid of the restriction of \(f\) to the subspace \(H\).
We will now formulate an analogue of Problem 2 for \(s\)-concave functions.
Problem 4. Let \(f \in C_s({\mathbb{R}}^n)\) with \(-1/(n+1) <s<\infty\), and let \(\nu(f)\) be the number of \((n-1)\)-dimensional affine subspaces \(H\) of \(\mathbb{R}^n\) passing through \(c(f)\) such that \(c_H(f)=c(f)\). What is the value of \(\nu_s(n) =\displaystyle \min_{f\in \mathcal{C}_s({\mathbb{R}}^n)} \nu(f)\)?
We will first show that \(\eta (2)=3\). The proof is similar to that for the Euclidean case; see e.g., [5]. To show that \(\eta(2)\le 3\), consider an equilateral Euclidean triangle \(\Delta\) in \(\mathbb{B}^2\) with centroid at the origin. Observe that \(\Delta\) is also convex in the hyperbolic sense, and the hyperbolic centroid of \(\Delta\) is at the origin, since the integral \[\int_{\Delta}\frac{p}{(1-|p|^2)^{3}}\,dp\] is invariant under rotations by \(2\pi/3\) with respect to the origin.
Note that a chord whose hyperbolic length is bisected by the origin is precisely a chord whose Euclidean length is bisected by the origin. Since, in the Euclidean setting, the origin bisects exactly three chords of \(\Delta\), the same holds in the hyperbolic setting.
Now consider an arbitrary convex body \(K\subset \mathbb{B}^2\) with centroid at the origin, i.e., \[\int_{K}\frac{p}{(1-|p|^2)^{3}}\,dp=0.\] Passing to polar coordinates, we obtain \[\int_0^{2\pi} \Psi(\rho(\varphi)) \cos(\varphi) d\varphi = 0 \quad \text{and} \quad \int_0^{2\pi} \Psi(\rho(\varphi)) \sin(\varphi) d\varphi = 0,\] where \[\Psi (s) = \int_0^s\frac{r^2}{(1-r^2)^3}\, dr,\] and \(\rho(\varphi)\) is the radial function of \(K\) in polar coordinates.
Using the same argument as in [5], one can show that the function \[\Psi(\rho(\varphi)) - \Psi(\rho(\varphi+\pi))\] has at least three roots in the interval \([0,\pi)\), which means that \(\rho(\varphi) = \rho(\varphi+\pi)\) for at least three values of \(\varphi\in [0,\pi)\). Thus, at least three chords of \(K\) are bisected by the origin, i.e., \(\eta(2)\ge 3\). Recalling that \(\eta(\Delta)= 3\), we obtain \(\eta(2)=3\).
We will now present the main result of this section. The proof is based on the Fourier transform of distributions. The reader is referred to [23] and [24] for background information.
Theorem 5. There exists a convex body \(K \subset \mathbb{B}^n\), \(n \geq 3\), with centroid at the origin, such that \[C(K \cap \xi^{\perp})\in\{x\in \mathbb{B}^n: x_n>0\}\] for all \(\xi \ne \pm e_n\).
Proof. Our goal is to construct a body \(K\), with centroid at the origin such that \[\begin{align} \boldsymbol{Z}_n(K\cap\xi_{\mathbb{B}}^\perp) &= \int_{K\cap\xi_{\mathbb{B}}^\perp}\frac{2^{n}\,p_n}{(1-|p|^2)^{n}}\,dp\label{eq:Zvecsec-unif} \end{align}\tag{14}\] is positive for all \(\xi \ne \pm e_n\).
As was shown in [25] (see the proof of Proposition 3.9), there exists an origin-symmetric convex body \(M\subset \mathbb{B}^n\) with strictly positive principal curvatures and \(C^\infty\) boundary such that \[\Phi(x)=\frac{\left\lVert x \right\rVert_M^{-1}}{1-\bigl(|x|/\left\lVert x \right\rVert_M\bigr)^2}\] is not a positive definite distribution on \(\mathbb{R}^n\).
Moreover, we can assume that \(M\) is rotationally invariant about the \(x_n\)-axis, and \(\widehat\Phi (e_n)<0\). Since \(\Phi\) is a homogeneous function of degree \(-1\) that is infinitely smooth on \(\mathbb{R}^n \setminus\{0\}\), \(\widehat\Phi\) is a homogeneous function of degree \(-n+1\) that is also infinitely smooth on \(\mathbb{R}^n \setminus\{0\}\); see [24].
Let \(\Omega(e_n)\) and \(\Omega(-e_n) \subset S^{n-1}\) be open spherical balls centered at \(e_n\) and \(-e_n\) respectively such that \(\widehat\Phi(\xi) < 0\) for all \(\xi \in \Omega(\pm e_n).\) Define an even function \(G \in C^{\infty}(S^{n-1})\) that is invariant under rotations about the \(x_n\)-axis and such that \[G(\xi) = \begin{cases} \text{positive}, \quad \xi \in \Omega(\pm e_n) \backslash \{\pm e_n\};\\ 0, \quad \xi \in \{\pm e_n\} \cup S^{n-1} \backslash \Omega(\pm e_n). \end{cases}\] By construction, \[\label{int95G} \int_{S^{n-1}}\widehat\Phi(\xi) G(\xi) d\xi < 0.\tag{15}\] Next, we define the function \(H \in C^{\infty}(S^{n-1})\) as \[H(x) = |x|^{-1} - (4(x_1^2 + \dots + x_{n-1}^2) + x_n^2)^{-\frac{1}{2}}.\] Note that \(H(x) > 0\) if \(x\in S^{n-1}\setminus\{\pm e_n\}\) and \(H(\pm e_n) = 0\). Extending \(H\) to \(\mathbb{R}^n\setminus\{ 0\}\) as a homogeneous function of degree \(-1\) and computing its Fourier transform using the well-known formulas
\[\left(|\cdot|^{-1} \right)^\wedge (x) = c_n |x|^{-n+1},\] and \[\left(|T y|^{-1} \right)^\wedge (x) = c_n |\det T|^{-1} |T^{-t}x|^{-n+1},\] where \(c_{n} = \frac{2^{n-1} \pi^{\frac{n}{2}} \Gamma\left(\frac{n-1}{2}\right)}{\Gamma\left(\frac{1}{2}\right)}\) and \(T\) is an invertible linear transformation on \(\mathbb{R}^n\), we obtain \[\begin{align} \widehat{H}(x) &=c_n \left(|x|^{-n+1} - \left(x_1^2 + \dots + x_{n-1}^2 + 4x_n^2\right)^{\frac{-n+1}{2}}\right). \end{align}\] Since \(\widehat{H} (x) \ge 0\) for \(x\in \mathbb{R}^n\setminus\{0\}\), an application of the spherical Parseval formula (see [24]) gives \[\label{int95H} \int_{S^{n-1}} \widehat\Phi(\xi) H(\xi) d\xi = \int_{S^{n-1}} \Phi(\xi) \widehat{H}(\xi) d\xi > 0.\tag{16}\] Now, for \(\lambda\in [0,1]\), we define \[g_\lambda (\xi) = (1-\lambda)G (\xi) + \lambda H (\xi), \qquad \xi\in S^{n-1}.\] Observe that \(g_\lambda\in C^\infty(S^{n-1})\), \(g_{\lambda}(\xi) > 0\) for all \(\xi \ne \pm e_n\), and \(g_\lambda(\pm e_n) = 0\).
Extending \(g_\lambda\) to \(\mathbb{R}^n\setminus\{0\}\) as a homogeneous function of degree \(-1\) and denoting the Fourier transform of this extension by \(\widehat{g_\lambda}\), we define a function \(\phi_\lambda\) on \(S^{n-1}\) by the formula \[\phi_\lambda(\xi) = \frac{1}{\xi_n} \widehat{g_\lambda}(\xi), \qquad \xi\in S^{n-1}.\] Note that \(\widehat{g_\lambda}(\xi)=0\) when \(\xi\in e_n^\perp\), and defining \(\phi_\lambda\) to be zero on \(e_n^\perp\) makes it a \(C^\infty\) function on \(S^{n-1}\) (for details see [5]). Also observe that \(\phi_\lambda\) is odd.
Consider the strictly increasing function \[\label{eq:def-Psi} \Psi(s) := \int_0^s \frac{r^{n-1}}{(1-r^2)^n}\,dr,\qquad s\in[0,1).\tag{17}\] For \(\lambda\in[0,1]\) and small enough \(\varepsilon>0\), we define the radial function \(\rho_{K_{\lambda,\varepsilon}}\) of a star body \(K_{\lambda,\varepsilon}\) by \[\label{eq:perturb} \Psi(\rho_{K_{\lambda,\varepsilon}}(\theta)) = \Psi(\rho_M(\theta))+\varepsilon\,\phi_\lambda(\theta), \qquad \theta\in S^{n-1}.\tag{18}\] Since \(M\) is convex with strictly positive principal curvatures, there is \(\varepsilon_\ast>0\) such that \(K_{\lambda,\varepsilon}\) is convex for all \(0<\varepsilon <\varepsilon_\ast\) and all \(\lambda\in[0,1]\). Also, \(\rho_{K_{\lambda,\varepsilon}}\) is rotationally invariant about the \(x_n\)-axis since \(\rho_M\) and \(\phi_\lambda\) are rotationally invariant and \(\Psi\) is strictly increasing.
For every \(\xi\in S^{n-1}\), we have \[\begin{align} \boldsymbol{Z}_n(K_{\lambda,\varepsilon}\cap\xi_{\mathbb{B}}^\perp) &= \int_{K_{\lambda,\varepsilon}\cap\xi_{\mathbb{B}}^\perp}\frac{2^{n}\,p_n}{(1-|p|^2)^{n}}\,dp \\ &= \int_{S^{n-1}\cap\xi^\perp}\int_0^{\rho_{K_{\lambda,\varepsilon}}(\theta)} r^{n-2} \frac{2^{n}r \, \theta_n}{(1-r^2)^{n}}\,dr\, d\theta\\ &= 2^n\int_{S^{n-1}\cap\xi^\perp} \Psi(\rho_{K_{\lambda,\varepsilon}}(\theta)) \, \theta_n \, d\theta\\ &= 2^n\int_{S^{n-1}\cap\xi^\perp} \left( \Psi(\rho_M(\theta))+\varepsilon\,\phi_\lambda(\theta)\right) \theta_n \, d\theta\\ &= 2^n \varepsilon\int_{S^{n-1}\cap\xi^\perp} \phi_\lambda(\theta) \, \theta_n \, d\theta\\ & = \frac{2^n \varepsilon}{\pi} \left(\phi_\lambda(x) x_n\right)^\wedge (\xi) = \frac{2^n \varepsilon}{\pi} \left(\widehat g_\lambda \right)^\wedge (\xi) = 4^n\pi^{n-1}\varepsilon g_{\lambda}(\xi)\ge 0. \end{align}\] Above we used the fact that if \(f\) is an even continuous function of degree \(-n+1\) on \(\mathbb{R}^n\setminus\{0\}\), then its Fourier transform is a homogeneous function of degree \(-1\) whose restriction to \(S^{n-1}\) equals \[\widehat f (\xi) = \pi \int_{S^{n-1}\cap \xi^\perp} f(\theta)\, d\theta,\qquad \xi\in S^{n-1},\] see [24].
We will now choose \(\lambda\) and \(\varepsilon\) so that the centroid of \(K_{\lambda,\varepsilon}\) is at the origin. First, we note that \[\begin{align} \boldsymbol{Z}_i(K_{\lambda,\varepsilon}) = \int_{K_{\lambda,\varepsilon}}\frac{2^{n+1}\,p_i}{(1-|p|^2)^{n+1}}\,dp = 2^{n+1} \int_{S^{n-1}}\int_0^{\rho_{K_{\lambda,\varepsilon}}(\theta)} \frac{r^n\, \theta_i}{(1-r^2)^{n+1}}\,dr\, d\theta =0 \end{align}\] for \(i = 1, \dots, n-1\), where we used that \(\rho_{K_{\lambda,\varepsilon}}\) is rotationally invariant about the \(x_n\)-axis. Therefore, it remains to show that \(\boldsymbol{Z}_n(K_{\lambda,\varepsilon}) = 0\). We have \[\begin{align} \boldsymbol{Z}_n(K_{\lambda,\varepsilon}) = \int_{K_{\lambda,\varepsilon}}\frac{2^{n+1}\,p_n}{(1-|p|^2)^{n+1}}\,dp = 2^{n+1} \int_{S^{n-1}}\int_0^{\rho_{K_{\lambda,\varepsilon}}(\theta)} \frac{r^n\, \theta_n}{(1-r^2)^{n+1}}\,dr\, d\theta. \end{align}\] Considering the latter as a function of \(\varepsilon\), we will obtain its expansion for small \(\varepsilon>0\). Using 18 with \(\varepsilon =0\), we have \[\Psi(\rho_{K_{\lambda,\varepsilon}}(\theta))\left.\right|_{\varepsilon=0} = \Psi(\rho_M(\theta)),i.e.,\rho_{K_{\lambda,\varepsilon}}(\theta)\left.\right|_{\varepsilon=0}=\rho_M(\theta),for all\theta\in S^{n-1}.\] Additionally, differentiating 18 , we get \[\left.\frac{d}{d\varepsilon} \rho_{K_{\lambda,\varepsilon}}(\theta) \right|_{\varepsilon=0}=\frac{\phi_{\lambda}(\theta)}{\Psi'(\rho_M(\theta))}=\frac{\phi_{\lambda}(\theta)(1-\rho_M^2(\theta))^n}{ \rho_M^{n-1}(\theta)}.\] Thus, \[\begin{align} \int_0^{\rho_{K_{\lambda,\varepsilon}}(\theta)}& \frac{ r^n }{(1-r^2)^{n+1}}\,dr \\ & = \int_0^{\rho_M(\theta)} \frac{ r^n }{(1-r^2)^{n+1}}\,dr + \varepsilon \frac{ \rho_M^n(\theta) }{(1-\rho_M^2(\theta))^{n+1}} \frac{\phi_{\lambda}(\theta)(1-\rho_M^2(\theta))^n}{ \rho_M^{n-1}(\theta)} +\varepsilon^2 R_{\lambda,\varepsilon} (\theta)\\ &= \int_0^{\rho_M(\theta)} \frac{ r^n }{(1-r^2)^{n+1}}\,dr + \varepsilon \frac{ \rho_M(\theta) }{1-\rho_M^2(\theta)} \phi_{\lambda}(\theta) +\varepsilon^2 R_{\lambda,\varepsilon} (\theta), \end{align}\] where the last term is the remainder in the Taylor expansion.
Since \(M\) is origin-symmetric, its radial function \(\rho_M\) is an even function on the sphere, and thus \[\begin{align} \int_{S^{n-1}}\int_0^{\rho_M(\theta)} \frac{r^n\, \theta_n}{(1-r^2)^{n+1}}\,dr\, d\theta=0. \end{align}\] Therefore, \[\begin{align} \boldsymbol{Z}_n({K_{\lambda,\varepsilon}}) & = 2^{n+1}\varepsilon \int_{S^{n-1}} \Phi(\theta) \theta_n\phi_{\lambda}(\theta) \, d\theta + \varepsilon^2 \bar R_{\lambda,\varepsilon}\\ & = 2^{n+1}\varepsilon \int_{S^{n-1}} \Phi(\theta) \widehat g_{\lambda}(\theta) \, d\theta + \varepsilon^2 \bar R_{\lambda,\varepsilon}\\ & = 2^{n+1}\varepsilon \int_{S^{n-1}} \widehat \Phi(\theta) g_{\lambda}(\theta) \, d\theta + \varepsilon^2 \bar R_{\lambda,\varepsilon}, \end{align}\] where \[\bar R_{\lambda,\varepsilon} = 2^{n+1}\int_{S^{n-1}} R_{\lambda,\varepsilon}(\theta)\, \theta_n \, d\theta.\] Note that \(\bar R_{\lambda,\varepsilon}\) is a continuous function of \(\varepsilon\) and \(\lambda\).
Define \[F(\lambda,\varepsilon)=\int_{S^{n-1}} \widehat\Phi(\xi) g_\lambda (\xi) \, d\xi+\varepsilon 2^{-n-1} \bar R_{\lambda,\varepsilon}.\]
Using 15 and 16 , we obtain \[F(0,0)=\int_{S^{n-1}} \widehat\Phi(\xi) G(\xi) \, d\xi<0\] and \[F(1,0)=\int_{S^{n-1}} \widehat\Phi(\xi) H(\xi) \, d\xi>0.\]
Since \(F(\lambda, \varepsilon)\) is a continuous map on \([0,1] \times [0, \varepsilon_\ast]\), there exists a small \(\varepsilon_0>0\) and \(\lambda_0 \in [0,1]\) such that \(F(\lambda_0, \varepsilon_0) = 0\), implying that the centroid of \(K_{\lambda_0,\varepsilon_0}\) is at the origin. ◻
Theorem 5 implies that \(\eta (n)\le 1\) for \(n\ge 3\). As we will see later in Remark 7, every convex body \(K\) in \(\mathbb{H}^n\), \(n\ge 3\), has at least one hyperplane section \(H\) passing through the centroid of \(K\) such that \(c(K\cap H) = c(K)\). Thus we obtain the following corollary.
Corollary 1. \(\eta (n)=1\) for \(n\ge 3\).
Remark 6. Using a similar construction for the sphere \(S^{n}\subset \mathbb{R}^{n+1}\), one can show that the spherical analogue of the number \(\eta(n)\) is equal to 1 for all \(n\ge 5\). However, as in the Euclidean space, our construction does not work in dimensions \(n=3\) and \(n=4\). Thus, we omit the details.
The following lemma is a functional analogue of the remark on p. 352 in [4]. One can also use topological methods as in [4], but we prefer to give a simple proof using analysis.
Lemma 1. Let \(f\in C_s(\mathbb{R}^n)\) with \(-1/(n+1)<s<\infty\) and \(0 \in \text{int}\left(\text{supp}(f)\right)\). Then there exists at least one direction \(u \in S^{n-1}\) such that \[\int_{u^\perp} x f(x) \, dx = 0.\]
Proof. Since an \(s_1\)-concave function is also \(s_2\)-concave for all \(s_2<s_1\), we can assume that \(f\) is \(s\)-concave with \(-1/(n+1)<s<0\). As was mentioned in the introduction, the integrals \[\int_{u^\perp} x f(x) \, dx\] are well defined for all \(u\in S^{n-1}\).
Consider the following function of \(u \in S^{n-1}\), \[F(u) = \int_{\{x: \, \langle x, u\rangle \geq 0\} } f(x) \, dx.\] \(F\) is continuous on \(S^{n-1}\) since \(f\) is integrable, and thus \(F\) attains its extreme values.
Fix \(u\in S^{n-1}\) and let \(v\) be a unit vector orthogonal to \(u\). For a real number \(\varphi\) close to zero, define \[u_v(\varphi) = \cos\varphi\, u +\sin \varphi\, v.\] We claim that \[\label{formula:der} \frac{d}{d\varphi} F(u_v(\varphi)) \Big|_{\varphi=0} = \int_{u^\perp} \langle y,v\rangle f(y) \, dy.\tag{19}\] Indeed, \[\begin{align} \label{eqn:deriv}\frac{d}{d\varphi} F(u_v(\varphi)) \Big|_{\varphi=0} &=\lim_{\varphi\to 0} \frac{1}{\varphi} \left(F(u_v(\varphi))- F(u)\right) \notag \\ &=\lim_{\varphi\to 0} \frac{1}{\varphi} \left( \int_{H^+} f(x) \, dx - \int_{H^-} f(x) \, dx\right) , \end{align}\tag{20}\] where \[H^+ = \{ x\in \mathbb{R}^n: \langle x , u\rangle \leq 0 \;\text{and}\;\langle x , u_v(\varphi)\rangle \geq 0 \}\] and \[H^- = \{ x\in \mathbb{R}^n: \langle x , u\rangle \geq 0 \;\text{and}\;\langle x , u_v(\varphi)\rangle \leq 0 \}.\] It is enough to compute the limit in 20 as \(\varphi\to 0^+\). The case \(\varphi\to 0^-\) will give the same result, since we can just replace \(\varphi\) with \(-\varphi\) and \(v\) with \(-v\).
First, we will evaluate \(\lim_{\varphi \to 0^+}\frac{1}{\varphi}\int_{H^+} f(x) \, dx\). Note that any point \(x \in H^+\) can be represented as \(x = y + t u\), where \(y \in u^\perp\), \(\langle y,v\rangle \ge 0\), and \(t \in [-\tan\varphi \langle y,v\rangle, 0]\). Then \[\begin{align} \lim_{\varphi\to 0^+} \frac{1}{\varphi} \int_{H^+} f(x) \, dx & = \lim_{\varphi\to 0^+} \frac{1}{\varphi}\int\limits_{\{ y\in u^\perp:\langle y,v\rangle\ge 0\}} \left( \int\limits_{-\tan\varphi \langle y,v\rangle }^0 f(y + tu)\, dt \right) \,dy\\ & = \lim_{\varphi\to 0^+} \frac{1}{\varphi}\int\limits_{\{ y\in u^\perp:\langle y,v\rangle\ge 0\}} \left(\int\limits_{-1 }^0 \tan\varphi \langle y,v\rangle f\left(y + \tan\varphi \langle y,v\rangle tu\right)\, dt \right)\, dy\\ & = \int\limits_{\{ y\in u^\perp:\langle y,v\rangle\ge 0\}} \left( \int\limits_{-1 }^0 \lim_{\varphi\to 0^+}\left( \frac{\tan\varphi}{\varphi} \langle y,v\rangle f\left(y + \tan\varphi \langle y,v\rangle tu\right)\right)\, dt \right)\, dy\\ & = \int\limits_{\{ y\in u^\perp:\langle y,v\rangle\ge 0\}} \left(\int\limits_{-1 }^0 \langle y,v\rangle f\left(y \right) \, dt \right)\, dy\\ & = \int\limits_{\{ y\in u^\perp:\langle y,v\rangle\ge 0\}} \langle y,v\rangle f\left(y \right) \, dy. \end{align}\] Above we used the Dominated Convergence Theorem to move the limit inside the integrals, since by 13 for small \(\varphi\) we have \[\begin{align} \left| \frac{\tan\varphi}{\varphi} \langle y,v\rangle f\left(y + \tan\varphi \langle y,v\rangle tu\right) \right| & \leq 2 |\langle y,v\rangle| f\left(y + \tan\varphi \langle y,v\rangle tu\right)\\ & \leq 2 C |\langle y,v\rangle| {(1+|y + \tan\varphi \langle y,v\rangle tu|^{-1/s})^{-1}}\\ & \le 2 C |\langle y,v\rangle| {(1+|y |^{-1/s})^{-1}}, \end{align}\] and the latter is an integrable function on \(u^\perp\).
The assumption that the origin lies in the interior of the support of \(f\) was used in computing the limit \[\lim_{\varphi\to 0^+}f\left(y + \tan\varphi \langle y,v\rangle tu\right)= f\left(y\right).\] Since \(f\) is continuous in the interior of its support, which is a convex set, the equality above holds for almost every \(y\in u^\perp\), except possibly on a set of zero \((n-1)\)-dimensional measure.
Thus, we obtain \[\lim_{\varphi\to 0^+} \frac{1}{\varphi} \int_{H^+} f(x) \, dx = \int\limits_{\{ y\in u^\perp:\langle y,v\rangle\ge 0\}} \langle y,v\rangle f(y)\, dy.\] Similarly, one can show \[\lim_{\varphi\to 0^+} \frac{1}{\varphi} \int_{H^-} f(x) \, dx = - \int\limits_{\{ y\in u^\perp:\langle y,v\rangle\le 0\}} \langle y,v\rangle f(y)\, dy.\] Formula 19 follows by subtracting the last two limits, and recalling that the limit as \(\varphi\to 0^-\) gives the same answer.
Let \(u_0 \in S^{n-1}\) be a direction, where \(F\) attains its minimum. Then, formula 19 implies \[\int_{u_0^\perp} \langle y,v\rangle f(y) \, dy=0,\] for any \(v\in u_0^\perp\), which yields the statement of the lemma. ◻
Remark 7. Lemma 1 remains valid if \(f\) is a continuous function supported on a star body. In particular if \[f(x) = \frac{1}{(1-|x|^2)^{n}}\] supported on a star body \(K\) that lies in the interior of \(\mathbb{B}^n\), then there is \(u\in S^{n-1}\) such that \[\int_{K\cap u^\perp}\frac{x}{(1-|x|^2)^{n}}\,dx=0.\] This is precisely what we needed for the proof of Corollary 1.
Proposition 8. There exists \(f\in C_s(\mathbb{R}^n)\) with \(-1/(n+1)<s<\infty\) and centroid at the origin such that \[\int_{u^\perp} x_n f(x) \, dx >0\] for all \(u\in S^{n-1}\) other than \(\pm e_n\).
Proof. Let \(g_s\) be an \(s\)-concave function defined as follows:
If \(-1/(n+1)<s<0\), \[g_s(x) = (1+|x|^2)^{1/s};\]
if \(s=0\), \[g_s(x)= e^{-|x|^2};\]
if \(s>0\), \[g_s(x) =\begin{cases} (9-|x|^2)^{1/s} , & |x|<3,\\ 0, & |x|\ge 3. \end{cases}\]
Define \[f(x) = g_s(x) + \varepsilon x_n b(|x|),\] where \(\varepsilon>0\) is sufficiently small and \(b\) is an infinitely smooth function on \(\mathbb{R}\) with support in the interval \([1,2]\) and such that \[\int_1^2 r^{n+1} b(r)\, dr = 0\] and \[\int_1^2 r^{n} b(r)\, dr > 0.\]
Since \(b\) has compact support, which is in the interior of the support of \(g_s\), and \(g_s\) is strictly \(s\)-concave, \(f\) is \(s\)-concave for small enough \(\varepsilon\).
Let us show that the centroid of \(f\) is at the origin. For each \(i=1,\ldots, n\), the function \(x_i g_s(x)\) is odd, and therefore \[\begin{align} \int_{\mathbb{R}^n} x_i f(x)\, dx&= \int_{\mathbb{R}^n} x_i \left( g_s(x) + \varepsilon x_n b(|x|) \right)\, dx= \varepsilon \int_{\mathbb{R}^n} x_i x_n b(|x|) \, dx\\ &= \varepsilon \int_{S^{n-1} } \theta_i \theta_n\, d\theta \int_1^2 r^{n+1} b(r) \, dr=0. \end{align}\]
For the restriction of \(f\) to the subspace \(u^\perp\), we have \[\begin{align} \int_{u^\perp} x_n f(x)\, dx&= \int_{u^\perp} x_n \left( g_s(x) + \varepsilon x_n b(|x|) \right)\, dx= \varepsilon \int_{u^\perp} x_n^2 b(|x|) \, dx\\ &= \varepsilon \int_{S^{n-1}\cap u^\perp } \theta_n^2 \, d\theta \int_1^2 r^{n} b(r) \, dr>0, \end{align}\] if \(u\ne \pm e_n\). ◻
Using Lemma 1 and Proposition 8 we obtain the following.
Corollary 2. Let \(-1/(n+1)<s<\infty\). Then \(\nu_s (n)=1\) for \(n\ge 2\).
Acknowledgment. We would like to thank Roman Vershynin for suggesting to study functional analogues of Problems 1 and 2.
All authors were supported in part by NSERC↩︎