1 Introduction and main results↩︎

Stability analysis investigates how close an object that almost attains the optimum is to an actual optimizer if one exists. Over the past two decades, this field has gained much attention and is nowadays an integral part in the study of functional and geometric inequalities; see [1]–[3], for instance. Among the latter, the isoperimetric inequality has been of primary interest in this context; see e.g. [4]–[7]. This inequality resolves the isoperimetric problem, which can be traced back to the legend of the Carthaginian queen Dido and asks for the maximal volume that can be enclosed by a given area; see e.g. [8], [9].

A first instance of a stability result in this direction is an inequality of Bonnesen [4] in dimension two; see also [10]. In higher dimensions, it took several decades until an optimal control measured in terms of the volume deviation from balls was obtained by Fusco, Maggi, and Pratelli in [7] via a symmetrization argument. An alternative proof via optimal transport that also applies in anisotropic settings was provided by Figalli, Maggi, and Pratelli shortly afterwards [11]; see also [12] for yet another versatile proof technique. Figalli and Indrei [13] used the approach in [11] to establish a quantitative version of a relative isoperimetric inequality inside an open convex cone \(\mathcal{C} \subset \mathbb{R}^n\), \(n\geq 2\), that does not contain lines due to Lions and Pacella [14]. More precisely, their result says that there exists a constant \(c(n,\mathcal{C})>0\) depending on the dimension \(n\) and on the cone \(\mathcal{C}\) such that if \(E\subset \mathcal{C}\) is a Borel set of finite perimeter and finite positive volume, then \[\left(\frac{\mathrm{Vol}(E\Delta (B_r^{Eucl}(0) \cap \mathcal{C}))}{\mathrm{Vol}(E)}\right)^2 \leq c(n,\mathcal{C}) \cdot \left(\frac{\mathrm{Area}(\partial E)}{ n \mathrm{Vol}(B_1^{Eucl}(0) \cap \mathcal{C})^{1/n} \mathrm{Vol}(E)^{\frac{n-1}{n}}}-1 \right),\] where \(r>0\) is such that \(\mathrm{Vol}(B_r^{Eucl}(0) \cap \mathcal{C})= \mathrm{Vol}(E)\). Here the left side is the square of the deviation in volume known as Fraenkel asymmetry and the term on the left side, which is always nonnegative by [14], is called the (relative) isoperimetric deficit. Such quantitative estimates have various applications. For instance, in the context of Gamow’s liquid drop model [15] from nuclear physics stability of the isoperimetric inequality is used to prove existence of ground states; see [16], [17].

Isoperimetric-type bounds are relevant in mathematical general relativity as well; see e.g. [18]–[22]. However, in Lorentzian signature one can neither estimate from above the volume of spacetime regions [23] nor the area of spacelike hypersurfaces with fixed boundary in terms of boundary area; see [24] though. In fact, minimization problems in Riemannian geometry rather correspond to maximization problems in Lorentzian geometry and vice versa. For instance, geodesics locally minimize arclength in the Riemannian case while they locally maximize it in the Lorentzian case. This reverse behavior is fundamental to special relativity and is reflected in phenomena such as the twin paradox. The main feature of Minkowski space that underlies this paradox is the reverse triangle inequality; see 7 . Similarly, one can prove reverse isoperimetric-type inequalities in Lorentzian signature, which can be applied to questions about black holes and cosmology as we explain in more detail towards the end of the introduction.

The first result in this direction is due to Bahn and Ehrlich [25], who obtained a Lorentzian analogue of (a special case of) the inequality of Lions and Pacella [14]: The area \(A(S)\) of a compact connected smooth spacelike achronal (hence acausal) hypersurface \(S\) with (piecewise) smooth boundary in the chronological future \(I^+(O)\) of the origin \(O\) in standard \((n+1)\)-dimensional Minkowski space \(\mathbb{L}^{n+1}\), \(n\in\mathbb{N}\), is bounded from above in terms of the volume \(V(C(S))\) of the past cone \(C(S)=\{tv \mid t\in [0,1],\,v\in S\}\). More precisely, in terms of its isoperimetric deficit, their inequality can be stated as \[\label{eq:BE95ineq} \delta_{BE} (S)\mathrel{\vcenter{:}}= (n+1) V(C(\pi(S)))^{\frac{1}{n+1}}\frac{V(C(S)) ^{\frac{n}{n+1}}}{A(S)} - 1 \geq 0 \,,\tag{1}\] where \(\pi: S\to \mathbb{H}\) denotes the radial projection to the (spacelike upper unit) hyperboloid \(\mathbb{H}\) centered at \(O\). For definitions of the area and volume, in the generality needed in this article, we refer to Subsection 3.2. Generalizations of the Bahn–Ehrlich inequality to other spacetimes are studied in [26]–[28].

The Bahn–Ehrlich inequality 1 is moreover sharp and rigid. Equality is achieved if and only if the hypersurface \(S\) is contained in a (spacelike upper) hyperboloid \(\mathbb{H}_t\) of some radius \(t>0\) centered at \(O\). In particular, like the reverse triangle inequality [29], the Bahn–Ehrlich inequality qualifies for a stability analysis in terms of its isoperimetric deficit 1 . We introduce the Fraenkel asymmetry of \(S\) as \[\label{eq:Frae} A_F(S) \mathrel{\vcenter{:}}= \frac{V( C(S) \Delta B_{t}(\mathbb{R}_+S))}{V(C(S))}\, ,\tag{2}\] where \(B_t(\mathbb{R}_+S)\) is the intersection of the cone \(\mathbb{R}_+ S\) with the past of \(\mathbb{H}_t\), and where the value \(t>0\) is chosen such that \(V(B_t(\mathbb{R}_+S))=V(C(S))\). A discussion on the Fraenkel asymmetry and its properties can be found in Subsection 4.3.

We call a cone \(M= \mathbb{R}_+ \Omega\) over a (nonempty) domain \(\Omega \subset \mathbb{H}\) with Lipschitz boundary a conical Minkowski spacetime. With its inherited Lorentzian structure, \(M\) is a special Friedmann-Lemaître-Robertson-Walker spacetime; see Subsection 2.2. Here and in the following, we refer to Section 2 for more details and complete definitions.

We can now state our first main result – a stability estimate for the Bahn–Ehrlich inequality for a larger set of admissible hypersurfaces \(S\). Here \(\mu\) denotes the measure on \(\mathbb{H}\) induced by its natural hyperbolic metric; see Subsection 2.1.

Note that, in contrast to [13], the constant in our stability estimate depends only on the dimension and not on an ambient cone.

The original proof by Bahn and Ehrlich of their Lorentzian isoperimetric inequality is based on a (Lorentzian) Brunn–Minkowski-type inequality, which is also the case in [14]. In Subsection 3.3 and 4.1, we provide simpler direct proofs of the isoperimetric Bahn–Ehrlich inequality – two functional-analytic ones and a geometric one, respectively. While all proofs might, in principle, serve as a starting point for a proof of Theorem 1, we quantify the geometric one, whose inductive step is based on an argument that appears in the equality discussion in [25]. Complementary stability results obtained by refining the functional-analytic proofs are discussed in Appendix 6. Nevertheless, the approach in those proofs to represent the hypersurface as a graph over a subset of the hyperboloid \(\mathbb{H}\) enters our proof of Theorem 1 as well. This idea leads us to a situation similar to the Euclidean case considered by Fuglede [5] with only nearly spherical sets as competitors. However, in contrast to [5], we do not impose a uniform \(W^{1,\infty}\)-bound, so we are not restricted to “nearly hyperbolic” hypersurfaces in that sense.

In this functional-analytic setting, up to a reduction argument to the compact setting in Subsection 4.4, our proof of Theorem 1 is given in Subsection 4.5. It relies on a decomposition into a sub- and a superlevel set, which boils down the proof to a quantitative Minkowski-type inequality; see Subsection 4.2. Here we exploit that, unlike in the Euclidean case, in our Lorentzian setting the surface area and the spacetime volume both behave additively under decompositions. The proof uses a slightly different but equivalent notion of Fraenkel asymmetry, which is discussed in Subsection 4.3. In Section 5 we give an example showing that in the theorem, like in the Euclidean case, the exponent \(2\) of the Fraenkel asymmetry is optimal, even for hypersurfaces in a fixed smooth conical Minkowski spacetime.

A different isoperimetric-type inequality for acausal hypersurfaces has recently been obtained by Cavalletti and Mondino [23] in the much more general setting of Lorentzian pre-length spaces [33] satisfying timelike Ricci curvature lower bounds in a synthetic sense via optimal transport [34]. As opposed to the Bahn–Ehrlich inequality [23] their approach has the advantage that it provides an isoperimetric-type inequality for the spacetime volume bounded by two nontrivial hypersurfaces. In the special setting of the Bahn–Ehrlich result above with only one nontrivial compact acausal hypersurface \(S\), their inequality can be stated in terms of its isoperimetric deficit as \[\label{eq:CM95ineq} \delta_{CM} (S)\mathrel{\vcenter{:}}= (n+1)\frac{ V(C(S))}{A(S)\cdot \operatorname{dist}(O,S)}-1\geq 0\, ,\tag{3}\] where \(\operatorname{dist}(O,S)\mathrel{\vcenter{:}}= \inf_{v\in S} |v|\) and \(|v|\) is the Lorentzian distance from \(O\) to \(v\); see Subsection 2.1. Also the Cavalletti–Mondino inequality is sharp and rigid. As before equality in 3 is attained if and only if the hypersurface \(S\) is contained in \(\mathbb{H}_t\) for some \(t>0\).

In the following proposition, we relate the Bahn–Ehrlich (BE) inequality 1 and the special Cavalletti–Mondino (CM) inequality 3 in terms of the relative volume excess \[\label{eq:RVE} \mathcal{E}(S)\mathrel{\vcenter{:}}= \frac{1}{n+1}\frac{V\left(C(S)\right)-V(B_{\operatorname{dist}(O,S)}(\mathbb{R}_+S))}{V(C(S))}\,,\tag{4}\] which satisfies \((n+1)\mathcal{E}(S)\in [0,1]\). Here the dimensional constant in \(\mathcal{E}(S)\) is included to lighten the presentation.

This observation on the relation between isoperimetric deficits is a simple consequence of Bernoulli’s inequality in the form \(1-(n+1)\mathcal{E}(S)\leq (1+\mathcal{E}(S))^{-(n+1)}\) and is useful to prove stability of related Lorentzian isoperimetric inequalities. Indeed, we deduce the following stability result for the CM-inequality from Proposition 3 by combining a simple consequence of Lemma 11, namely, \[\label{eq:RVEtoFr} A_F(S)\leq 2(n+1)\mathcal{E}(S)\,\tag{5}\] with the BE-inequality in the generality as stated in Theorem 1.

While the BE-inequality 1 demonstrates the same stability behavior with a power \(2\) of the Fraenkel asymmetry \(A_F\) as the Euclidean isoperimetric inequality [7] and other geometric inequalities, e.g. [13], [35], [36], the CM-inequality 3 experiences a strengthened stability behavior. Strong stability results with a power of the distance equal to \(1\) – as found in Corollary 4 – seem to be quite rare in the literature; see the two-dimensional result in [37], though, and Subsection 1.1 for a comparison with the functional setting.

As another consequence of Proposition 3 and the BE-inequality, we obtain an improved isoperimetric-type inequality, which can be expressed in terms of a new isoperimetric deficit \[\label{eq:deltaCMstar} \delta_{CM}^*(S)\mathrel{\vcenter{:}}= \delta_{CM}(S)-\mathcal{E}(S) \geq 0\,.\tag{6}\] By Proposition 3 the isoperimetric deficits are related as \[\delta_{CM}(S)\geq \delta_{CM}^*(S)\geq \delta_{BE}(S) \geq 0\,.\] In particular, sharpness of the CM-inequality implies sharpness of the refined CM-inequality, and rigidity of the BE-inequality implies rigidity of the refined CM-inequality.

Moreover, a consequence of Theorem 1 and Proposition 3 is that the refined CM-inequality, as opposed to the CM-inequality, exhibits the usual quadratic stability behavior.

In Appendix 7 we provide an alternative proof of this quantitative refined CM-inequality that is independent of Theorem 1 and gives an improved stability constant. Observe also that applying 5 turns Corollary 5 into a higher-order stability estimate of Corollary 4.

As consequence of their isoperimetric inequality, Cavalletti and Mondino discuss in [23] upper bounds on the area of acausal hypersurfaces inside the interior of a black hole and in cosmological spacetimes. In the latter case they work in a conical spacetime and interpret the tip of the cone, which corresponds to the origin \(O\) in our special setting, as “big bang”, and the distance \(\operatorname{dist}(O,S)\) roughly as the “age” of the universe. In the special case of a conical Minkowski spacetime, Corollary 4 and 5 yield improved upper bounds.

1.1 Comparison with the stability of functional inequalities↩︎

We complement the discussion on stability exponents for isoperimetric-type inequalities with results for functional inequalities, where this topic has attracted much attention in the past few years. Indeed, while in the seminal work [38] a quadratic stability result in terms of the gradient \(L^2\)-norm as notion of distance to the set of optimizers has been established, (optimal) non-quadratic results were obtained only very recently; see, for instance, [39]–[41] for quartic stability results caused by degeneracy and [42]–[45] for non-quadratic results in the absence of a natural Hilbert space structure. In contrast to these results, the quantitative CM-inequality has a stability exponent with respect to the Fraenkel asymmetry that is smaller than quadratic, which left scope for an improved stability result; see Corollary 5.

As mentioned before, Lorentzian inequalities have the reversed sign as compared to their Euclidean analogue. Reverse functional inequalities are prominent in the literature and arise mainly as lower dimensional extensions of well-known inequalities; see [46]–[49], for instance. In contrast, the sign-reversion of the Lorentzian inequalities is caused by the Lorentzian metric. Intriguingly, while explicit stability constants are usually rather difficult to obtain (see [50]), the (sharp) stability constant for the reverse Sobolev inequality can be found through a short and elegant argument [51], [52]. Similarly, the stability constants for our Lorentzian inequalities are explicit and follow (at least for the (refined) CM-inequality in Appendix 7) via an ad-hoc argument. However, we do not know yet whether they are (asymptotically) optimal.

1.2 Structure of the paper↩︎

In Section 2 we recall some basic facts about Lorentzian geometry and introduce important notions such as achronal hypersurfaces and our notions of area and volume.

In Section 3 we characterize achronal Lipschitz hypersurfaces in terms of graphs and express relevant quantities in functional form. This allows us to give two direct, alternative proofs of the BE-inequality.

Our main stability result for the BE-inequality is proved in Section 4. More specifically, after motivating crucial ideas in yet another, geometric proof of the BE-inequality, we establish the quadratic stability result with respect to the Fraenkel asymmetry as given in Theorem 1. A key ingredient is a Minkowski-type inequality in quantitative form. Sharpness of our stability results with respect to the Fraenkel asymmetry is discussed in Section 5.

Finally, the appendix supplements these results by providing complementary stability results of the BE-inequality and an ad-hoc, alternative proof of the refined CM-inequality.

Acknowledgements. The first named author would like to thank Bernhard Leeb for support that allowed him to attend talks by Alessio Figalli and Andrea Mondino on stability of geometric inequalities and on Lorentzian isoperimetric inequalities, respectively. The second named author would like to thank Rupert Frank for introducing him to the stability of functional inequalities and constant support. We moreover would like to thank Andrea Mondino for pointing out the work of Bahn and Ehrlich. Partial support through the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) grants FR 2664/3-1 and TRR 352-Project-ID 470903074 and through the Studienstiftung des deutschen Volkes (J.W.P.) is acknowledged.

2 Preliminaries↩︎

2.1 Some basic Lorentzian geometry↩︎

We recall some basics on Lorentzian geometry and refer, for instance, to [30] for more details. Let \(\mathbb{L}^{n+1}\), \(n\in \mathbb{N}\), be an \((n+1)\)-dimensional Minkowski vector space with Minkowski inner product \(\langle\cdot, \cdot \rangle\) of signature \(-+++\ldots\) and its standard topology. We denote the corresponding Lorentzian norm as \(|v| \mathrel{\vcenter{:}}= \sqrt{|\langle v , v \rangle|}\). A vector \(v\in \mathbb{L}^{n+1}\) is called spacelike if \(\langle v,v \rangle > 0\) and timelike if \(\langle v,v \rangle < 0\). We impose a time-orientation on \(\mathbb{L}^{n+1}\) by choosing a timelike vector \(v_0 \in \mathbb{L}^{n+1}\) and calling a non-spacelike vector \(v\) future-directed if \(\langle v,v_0 \rangle < 0\). The chronological future of the origin \(O\) in \(\mathbb{L}^{n+1}\) is the set of future-directed timelike vectors in \(\mathbb{L}^{n+1}\) and is denoted by \(I^+(O)\). We note that every element in \(I^+(O)\) induces the same time orientation on \(\mathbb{L}^{n+1}\). Every pair of vectors \(u,v \in I^+(O)\) satisfies the reverse triangle inequality \[\label{eq:reverse:triangle} |u| + |v| \leq |u+v| \, .\tag{7}\] For a subset \(A\subset I^+(O)\) we write \[\operatorname{dist}(O,A)\mathrel{\vcenter{:}}= \inf_{v \in A} |v| \, .\]

A nondegenerate bilinear form on an oriented vector space induces a volume form on this vector space. In our setting, if \(e_0,\ldots,e_{n}\) is an oriented orthonormal basis of \((\mathbb{L}^{n+1},\langle\cdot, \cdot \rangle)\), i.e. \(\langle e_j,e_k \rangle \in \{ \pm \delta_{jk}\}\), then this volume form is given by \[\omega = e_0^* \wedge \ldots \wedge e_n^*\,,\] where \(e_0^*,\ldots,e_{n}^*\) is a dual basis of \(e_0,\ldots,e_{n}\). The volume form on \(\mathbb{L}^{n+1}\) in turn induces a measure \(\mu_{\mathbb{L}}\), which is independent of the chosen orientation. We denote the measure \(\mu_{\mathbb{L}}(B)\) of a measurable subset \(B\) of \(\mathbb{L}^{n+1}\) by \(V(B)\) and call it the volume of \(B\). This volume coincides with the volume induced by the Euclidean inner product \(\langle\cdot, \cdot \rangle_{\mathrm{Eucl}}\) defined by \(\langle e_j,e_k \rangle_{\mathrm{Eucl}} = \delta_{jk}\); cf. [25].

By a hypersurface of \(\mathbb{L}^{n+1}\) we mean a codimension one \(\mathcal{C}^{0}\)-submanifold of \(\mathbb{L}^{n+1}\) with boundary. We call such a hypersurface Lipschitz if the submanifold with boundary is of class \(\mathcal{C}^{0,1}\). As we are going to work primarily with Lipschitz continuous objects, we refer the reader to [53] for some background material. A Lipschitz hypersurface is called spacelike (non-timelike), if almost all of its tangent spaces contain only spacelike (non-timelike) nontrivial vectors.

The volume form \(\omega\) on \(\mathbb{L}^{n+1}\) and the chosen time orientation induce an almost everywhere defined, locally uniformly bounded family of alternating \(n\)-forms on any non-timelike Lipschitz hypersurface \(S\) of \(\mathbb{L}^{n+1}\), which, in turn, induces a measure \(\mu_S\) on \(S\) via integration that is independent of the chosen time orientation. We denote the total measure \(A(S)\mathrel{\vcenter{:}}= \mu_S(S)\) and call it the area of \(S\). Using the notions of \(V\) and \(A\) above, the definitions of the deficits 1 , 3 , and 6 as well as of the Fraenkel asymmetry 2 from the introduction extend naturally to the generality required for Theorem 1.

Of particular interest to us are the (upper) hyperboloids of radius \(r>0\) given by \[\mathbb{H}_r \mathrel{\vcenter{:}}= \{v \in \mathbb{L}^{n+1} \mid \langle v,v\rangle = -r^2, \langle v,v_0\rangle < 0\} \subset I^+(O) \, ,\] all of which are smooth spacelike hypersurfaces. The Minkowski inner product of \(\mathbb{L}^{n+1}\) restricts to a smooth Riemannian metric on \(\mathbb{H}_r\) with constant negative sectional curvature \(-1/r^2\). In particular, \(\mathbb{H}\mathrel{\vcenter{:}}=\mathbb{H}_1\) is isometric to the \(n\)-dimensional hyperbolic space. We write \(\pi: I^+(O) \rightarrow \mathbb{H}\) for the radial projection, \(d_{\mathbb{H}}\) for the induced metric on \(\mathbb{H}\), and \(\mu\) for the induced measure on \(\mathbb{H}\).

2.2 Conical Minkowski spacetimes and Cauchy hypersurfaces↩︎

By a domain in \(\mathbb{H}\) we mean a nonempty \(n\)-dimensional Lipschitz submanifold of \(\mathbb{H}\) with boundary. In particular, it has nonempty interior. While a domain is often supposed to be connected, our results all hold without assuming connectedness. By a conical Minkowski spacetime we mean a cone \(M=\mathbb{R}_{+} \Omega\) for some domain \(\Omega \subset \mathbb{H}\). We note that \(\mathbb{L}^{n+1}\) induces the structure of a time-oriented Lorentzian manifold with boundary on \(M\). We further note that \(M\) is convex if and only if \(\Omega\) is convex. A conical Minkowski \(M\) spacetime can be represented as a Lorentzian warped product of an interval and its defining domain \(\pi(M)\); see [54]. As described in the introduction, we define \(B_t(M)\mathrel{\vcenter{:}}= C(\mathbb{H}_t\cap M)\backslash \mathbb{H}_t\), \(t>0\).

A locally Lipschitz continuous curve \(\gamma:I\rightarrow \mathbb{L}^{n+1}\) on some interval \(I\) is called non-spacelike or timelike if \(\gamma'(t)\) satisfies the respective property for almost every \(t\in I\). A non-spacelike curve \(\gamma\) is also called causal. It is called future-directed if \(\langle \gamma'(t),v_0 \rangle < 0\) holds in addition almost everywhere. The Lorentzian length of a causal curve is defined as \[L(\gamma)=\int_I | \gamma'(t) | \, \mathrm dt \, .\]

A subset of a conical Minkowski spacetime \(M\) is called achronal resp. acausal (in \(M\)) if any timelike resp. non-spacelike curve contained in \(M\) intersects it at most once. It is called a Cauchy hypersurface if it is intersected exactly once by every timelike curve. Clearly, any acausal set in \(M\), any Cauchy hypersurface of \(M\), and any achronal set in \(I^+(O)\) that is contained in \(M\) is also achronal in \(M\). In particular, in the proofs of our results we can always assume that \(M=\mathbb{R}_+ S\). The converses do however not hold in general. Any achronal Lipschitz hypersurface is non-timelike; see Lemma 7. There a description of all achronal sets in a conical Minkowski spacetime can be found as well.

2.3 Induced length metrics↩︎

Let \((X,d)\) be a metric space. We call it extended if the metric \(d\) is allowed to take the value \(\infty\). The length of a (continuous) curve \(\gamma : [a,b] \rightarrow X\) is defined as \[L_d(\gamma) \mathrel{\vcenter{:}}= \sup \left\{ \sum_{i=0}^{N-1} d(\gamma(t_i),\gamma(t_{i+1})) \Bigg{|} \,N \in \mathbb{N},\, a=t_0 < t_1 <\dots < t_N=b \right\} \, .\] A curve \(\gamma\) is called rectifiable if \(L_d(\gamma) < \infty\). The induced length metric on \(X\) is defined as \[d_L(x,y) \mathrel{\vcenter{:}}= \inf \left\{L_d(\gamma) \mid \gamma : [a,b] \rightarrow X \, \,\mathcal{C}^0\text{-curve with }\gamma(a)=x, \gamma(b)=y \right\} \geq d(x,y) \, .\] The pair \((X,d_L)\) is an extended metric space; see [55], for instance. The metric space \((X,d)\) is called intrinsic or a length space if \(d=d_L\).

Every rectifiable curve in a metric space can be parametrized by arclength; see [55]. With respect to this parametrization, it is a Lipschitz curve. In particular, in the definition of the induced length metric, we can equivalently work with the infimum over all Lipschitz curves from \(x\) to \(y\).

In an intrinsic metric space \((X,d)\), it is elementary to check that the following statements about a continuous function \(f: X \rightarrow \mathbb{R}\) are equivalent.

(i) \(f\) is \(L\)-Lipschitz continuous (on a dense subset)

(ii) \(f\) is locally \(L\)-Lipschitz continuous

(iii) \(|\nabla f(x_0)|\mathrel{\vcenter{:}}=\limsup_{x\rightarrow x_0} |f(x)-f(x_0)|/d(x,x_0) \leq L\) for all \(x_0 \in X\).

We denote the intrinsic metric of a domain \(\Omega \subset \mathbb{H}\) by \(d_\Omega\) and observe that the restrictions of \(d_\mathbb{H}\) and \(d_\Omega\) to a convex subset of \(\Omega\) coincide. In particular, we see that in the interior of such a domain \(\Omega\), the conditions (ii) and (iii) above can be equivalently stated with respect to either \(d_\mathbb{H}\) or \(d_\Omega\). We also note a locally Lipschitz continuous function \(f:\Omega \rightarrow \mathbb{R}\) is almost everywhere differentiable by Rademacher’s theorem and that the quantity in (iii) coincides with the norm of its gradient wherever it exists.

3 Functional representations and Bahn–Ehrlich isoperimetric inequality↩︎

In this section we consider a representation of achronal sets as graphs over the hyperboloid \(\mathbb{H}\). This provides the framework in which we prove Theorem 1; in particular, it allows us to obtain the BE-inequality for more general hypersurfaces.

3.1 Timelike curves↩︎

To prove this graphical characterization in Subsection 3.2, we first need to quantify the regions that are reachable by timelike curves.

3.2 Graphs and functional representations↩︎

It is well-known that an achronal set \(S \subset I^+(O)\) can be represented as the graph of a \(1\)-Lipschitz function \(h:Z \to \mathbb{R}\) for some subset \(Z \subset \mathbb{R}^n\); cf. [56] and [31]. Moreover, away from its edge such an achronal set is known to be a hypersurface with empty boundary [30], and hence Lipschitz; cf. the next lemma. Similarly, we can represent such an achronal set \(S\) as a graph over the subset \(\pi(S)\) of the hyperboloid \(\mathbb{H}\). Recall that \(|\cdot|\) denotes the Minkowski norm in \(\mathbb{L}^{n+1}\) and the hyperbolic norm in case of a tangent vector of \(\mathbb{H}\), respectively.

Now we want to express the area \(A(S)\) and the volume \(V(C(S))\) for an achronal Lipschitz hypersurface \(S=S_f\) in terms of the function \(f\) provided by Lemma 7. To this end, we equip \(\mathbb{R}_+ \times \mathbb{H}\) with the product measure of the Lebesgue measure \(\lambda^1\) on \(\mathbb{R}_+\) and the hyberbolic measure \(\mu\) on \(\mathbb{H}\). The diffeomorphism \(\Phi: \mathbb{R}_+ \times \mathbb{H}\rightarrow I^+(O)\) defined by \(\Phi(r,x)=rx\) satisfies \(\mathrm d\Phi_{(r,x)} (1,y)=ry+ x\). Hence, \((\Phi^{-1})_* \mu_{\mathbb{L}} = r^{n}\lambda^1\otimes \mu\). Then we find that \[\label{eq:volume95formula} V(C(S))= \int_{\pi(S)} \int_0^{f(x)} r^{n}\,\mathrm d\lambda^1(r)\,\mathrm d\mu(x) = \frac{1}{n+1}\int_{\pi(S)} f(x)^{n+1} \,\mathrm d\mu(x)\,.\tag{9}\] In the remainder of this article, we suppress the Lebesgue measure in the notation and just write \(\mathrm d t\), for instance.

Note that if \(\pi(S)\) is a ball of radius \(t^*>0\) in \(\mathbb{H}\) and the function \(f\) is rotationally symmetric around the center \(x_0\) of this ball, i.e. \(f(x)=r(d_{\mathbb{H}}(x_0,x))\), then the integral representation for \(V(C(S))\) disintegrates to \[\label{eq:volume95radial} V(C(S))=\frac{1}{n+1}\int_{0}^{t^*} r(t)^{n+1} \,\mathrm d\nu(t)\,,\tag{10}\] where \(\mathrm d\nu(t)=nV(B_1^{Eucl})\sinh^{n-1}(t)\,\mathrm d t\) and, as before, \(B_1^{Eucl}\) denotes the \(n\)-dimensional Euclidean unit ball.

Next we turn to the area \(A(S)\) of \(S=S_f\). As the function \(f\) is in general only Lipschitz continuous, all pointwise computations that are about to follow and involve the derivative of \(f\) are to be understood in the almost everywhere sense. Set \(F: \pi(S) \to \mathbb{L}^{n+1}\), \(F(x)=f(x) x\). For a point \(x\in \pi(S)\) and an orthonormal basis \(v_1,\ldots,v_n\) of \(T_x \pi(S)\) we have \[\mathrm dF_x v_j = \partial_{v_j}f (x) x + f(x) v_j\] and \[\langle \mathrm dF_x v_j, \mathrm dF_x v_k \rangle = -\partial_{v_j} f(x) \partial_{v_k}f(x) + f(x)^2 \delta_{jk}\,.\] The determinant of this Gram-matrix is given by \[f(x)^{n-1} \sqrt{f(x)^2 - |\nabla f (x)|^2}= f(x)^n \sqrt{1- |\nabla \ln(f) (x)|^2}\,.\] Note that the expression under the square root is nonnegative by Lemma 7, (i). Hence, we have \[\label{eq:area95formula} A(S) = \int_{\pi(S)} f(x)^n \sqrt{1- |\nabla \ln(f) (x)|^2} \,\mathrm{d} \mu(x)\,.\tag{11}\] In particular, in the rotationally symmetric case, we have \[\label{eq:area95radial} A(S) \mathrel{\vcenter{:}}= \int_{0}^{t^*} r(t)^{n-1}\sqrt{ r(t)^2-r'(t)^2}\,\mathrm d\nu(t)=\int_{0}^{t^*} r(t)^{n}\sqrt{ 1- (\ln(r)'(t))^2}\,\mathrm d\nu(t)\,.\tag{12}\] In the following, we also write \(A(B)=\mu_S(B)\) for a general \(\mu_S\)-measurable subset \(B\) of \(S\) and note that the above formulas extend naturally to this case. This turns out to be useful in the proof of Theorem 1. More specifically, it enters a generalization of the Bahn–Ehrlich inequality, which we prove in the next subsection.

3.3 Two functional-analytic proofs of the Bahn–Ehrlich inequality↩︎

As a first application of the discussed representation \(S=S_f\), \(f: \mathbb{H}\supset \pi(S) \rightarrow \mathbb{R}\), of an achronal hypersurface \(S\) provided by Lemma 7, we give two direct proofs for the BE-inequality without relying on a Lorentzian Brunn-Minkowski inequality or approximation arguments but based on Hölder’s inequality and Bernoulli’s inequality, respectively. This approach moreover shows that the BE-inequality holds for general \(\mu_S\)-measurable subsets of an achronal hypersurface, which will be important for our stability proof in Subsection 4.5.

Note that, in the situation of the theorem, \(B\) is \(\mu_S\)-measurable in \(S\) if and only if \(\pi(B)\) is \(\mu\)-measurable as the projection \(\pi_S : S \rightarrow \pi(S)\) is a homeomorphism.

In both proofs, the function \(f\) is such that \(S=S_f\) as in Lemma 7, and we set \(g=f^{n+1}\).

This proof was based on Hölder’s inequality. Under the slightly stronger technical assumption \(V(C(B)),V(C(\pi(B)))\in \mathbb{R}_+\), we obtain another proof using Bernoulli’s inequality.

We refer to Appendix 6 for a stability discussion of the BE-inequality using quantitative versions of Hölder’s or Bernoulli’s inequality. Even though it is conceivable that they exhibit quantitative versions that imply stability of the BE-inequality with respect to the Fraenkel asymmetry, we pursue a different strategy and reduce the stability of the BE-inequality to a quantitative version of an inequality by Minkowski; see Subsection 4.2.

4 Stability of the Bahn–Ehrlich isoperimetric inequality↩︎

Our goal in this section is to prove Theorem 1. In the first subsection we provide further proofs of the BE- and CM-inequality and meanwhile discuss some crucial ingredients that we need in the proof of stability in Theorem 1 and Corollary 4.

4.1 A geometric proof of the Bahn–Ehrlich inequality↩︎

Bahn and Ehrlich [25] proved their isoperimetric-type inequality 1 as a corollary of a Lorentzian Brunn–Minkowski inequality. Here we show that it can also be deduced from simple geometric considerations on a spacelike \(n\)-dimensional simplex \(P\subset I^+(O)\).

Indeed, let \(\Pi\subset \mathbb{L}^{n+1}\) be the supporting hyperplane of \(P\) and let \(h=d_H(O,\Pi)\) be the Lorentzian height of \(C(P)\). Similarly to the Euclidean case, we have \[\label{eq:coneform} V(C(P)) = \frac{1}{n+1} hA(P)\,;\tag{13}\] see [25]. As \(\mathbb{H}_h\) is tangent to \(\Pi\), we observe that \(C(P)\subset B_h(M)\) for \(M= \mathbb{R}_+ P\), and thus \(V(C(P))\leq V(B_h(M))=h^{n+1}V(B_1(M))\). Note that a similar construction but with the supremum of \(B_t(M)\) over \(t>0\) contained in the past of the hypersurface gives the crucial quantity \(\mathcal{E}(S)\) defined in 4 and used for refining our BE-inequality; see Proposition 3. The obtained bound together with 13 implies that \[\begin{align} A(P) &= (n+1)h^{-1}V(C(P))^{\frac{1}{n+1}} V(C(P)) ^{\frac{n}{n+1}} \leq (n+1)V(B_1(M))^{\frac{1}{n+1}} V(C(P))^{\frac{n}{n+1}} \, , \end{align}\] which is nothing else but the BE-inequality for \(P\).

Its extension to general hypersurfaces then follows from induction and approximation as employed by Bahn and Ehrlich [25] for the Lorentzian Brunn–Minkowski inequality. More specifically, we decompose a polyhedral spacelike hypersurface \(S\subset I^+(O)\) into alike hypersurfaces \(S_1\) and \(S_2\) with less simplices and suppose that the BE-inequality holds for \(S_1\) and \(S_2\). Then, with \(\sigma_i = V(C(\pi(S_i)))\), \(i=1,2\), and \(\sigma = V(C(\pi(S))) = \sigma_1 + \sigma_2\), we have \[\begin{align} \notag A(S) &=A(S_1) + A(S_2) \\\notag &\leq (n+1) \left( \left(\sigma_{1} V(C(S_1))^n\right)^{\frac{1}{n+1}} + \left(\sigma_{2}V(C(S_2))^n\right)^{\frac{1}{n+1}} \right) \\\notag &\leq (n+1) \left((\sigma_{1}+\sigma_{2}) (V(C(S_1))+V(C(S_2)))^n\right)^{\frac{1}{n+1}} \\\label{eq:inductarg} & = (n+1) (\sigma V(C(S))^n)^{\frac{1}{n+1}}\,, \end{align}\tag{14}\] where we used an inequality of Minkowski [58] in the penultimate step. Hence, in the polyhedral case, the BE-inequality follows from the one for simplices above by induction. A quantitative version of this Minkowski inequality is studied in the next subsection. It will be an essential piece in our stability proof.

An extension to general achronal Lipschitz hypersurfaces can now be obtained by a standard approximation argument; cf. [25].

Using \(\operatorname{dist}(O,S)\leq h\), it turns out that, in a conical Minkowski spacetime, the CM-inequality 3 follows via a similar but simpler argument.

4.2 A quantitative Minkowski inequality↩︎

The first step in our proof towards a stability result for the Bahn–Ehrlich inequality in terms of the Fraenkel asymmetry is the following quantitative Jensen-type inequality.

Note that a second order Taylor expansion shows that the constant \(p(1-p)/8\) is best possible. For our purposes, though, the optimal constant is not essential.

As a consequence of Lemma 9 we obtain the following quantitative version of an inequality by Minkowski [58].

Before we discuss the reduction to compact sets and prove Theorem 1, we discuss the notion of Fraenkel asymmetry and its functional formulation used in the proof.

4.3 Notion of distance↩︎

Let \(S\) be an achronal Lipschitz hypersurface in a conical Minkowski spacetime \(M=\mathbb{R}_+S\) with \(V(C(S))<\infty\) and \(V(B_1(\mathbb{R}_+S))<\infty\). In the proof of Theorem 1 in Subsection 4.5, it turns out to be convenient to work with the asymmetry defined as \[\label{eq:AF95tilde} \tilde{A}_F(S)\mathrel{\vcenter{:}}=\inf_{t>0} \frac{V( C(S) \Delta B_t(\mathbb{R}_+S))}{V(C(S))}\,.\tag{15}\] It can be equivalently formulated in terms of functions as \[\tilde{A}_F(S) = \inf_{t>0} \frac{V( C(S) \Delta B_t(\mathbb{R}_+S))}{V(C(S))} = \frac{1}{(n+1)V(C(S))}\inf_{t>0}\int_{\pi(S)} |f(x)^{n+1}-t^{n+1}|\,\mathrm d\mu(x)\,,\] where \(f\) is such that \(S=S_f\) as in Lemma 7. Since this is a distance for the \((n+1)\)-th power of \(f\) in the \(L^1\)-norm, it can be regarded as an \(L^1\)-distance.

The asymmetry 15 is equivalent to the Fraenkel asymmetry 2 in the following sense.

4.4 Reduction to the compact case↩︎

In this subsection we approximate the (potentially non-compact) domain \(\Omega=\pi(S)\) by increasing compact domains \(\Omega_k\) and show that we can, without loss of generality, assume our achronal Lipschitz hypersurface in Theorem 1 to be compact.

Recall from Lemma 7, (iii), that compactness of an achronal Lipschitz hypersurface \(S\) is equivalent to compactness of its associated domain \(\pi(S)\).

Now we can finally prove our quantitative version of the Bahn–Ehrlich inequality.

4.5 Proof of Theorem 1↩︎

We work with the function \(f :\pi(S) \rightarrow \mathbb{R}_+\) that represents our achronal hypersurface \(S=S_f\), and we can assume that \(M=\mathbb{R}_+ S\) as before. By the approximation argument in Lemma 12, we can assume that \(\Omega=\pi(S)\) is compact.

We pick some \(t_0 \in \mathbb{R}_+\) such that \(\mu(\{f<t_0\})\) and \(\mu(\{f> t_0\})\) are not larger than \(\mu(\Omega)/2\). Next we decompose the level set \(\{f=t_0\}=E_1 \cup E_2\) into two \(\mu\)-measurable \(E_1\) and \(E_2\) such that \[\mu(\{f<t_0\})+\mu(E_1)=\mu(\{f>t_0\})+\mu(E_2) = \mu(\Omega)/2\,.\] To find such a decomposition, we consider a linear function \(\lambda : \mathbb{L}^{n+1} \rightarrow \mathbb{R}\) with spacelike level sets and observe that the map \(s \mapsto \mu (\{f=t_0\} \cap \{\lambda < s\})\) is continuous by boundedness of \(\Omega\) and the fact that intersections between \(\mathbb{H}\) and level sets of \(\lambda\) are \(\mu\)-null sets. We set \(\Omega_1=\{f<t_0\} \cup E_1\) and \(\Omega_2=\{f<t_0\} \cup E_2\). Then \(\Omega_1\) and \(\Omega_2\) are disjoint \(\mu\)-measurable sets with \(\Omega=\Omega_1\cup\Omega_2\), \(\mu(\Omega_1)=\mu(\Omega_2)=\mu(\Omega)/2\), and \(\sup_{\Omega_2} g \leq \inf_{\Omega_1} g\).

We set \(\sigma\mathrel{\vcenter{:}}= V(C(\Omega))\) and \(B_i=S \cap \mathbb{R}_+ \Omega_i\), \(i=1,2\). Then \(B_i\subset S \subset M\), \(i=1,2\), satisfy the assumptions of the BE-inequality in the version of Theorem 8. Hence, also the inductive argument given in 14 holds. More specifically, the BE-inequality and the previously introduced inequality of Minkowski imply \[\begin{align} A(S) &=A(B_1) + A(B_2) \leq (n+1) \left( \left(\sigma/2\, V(C(B_1))^n\right)^{\frac{1}{n+1}} + \left(\sigma/2\, V(C(B_2))^n\right)^{\frac{1}{n+1}} \right) \\ &\leq (n+1) (\sigma V(C(S))^n)^{\frac{1}{n+1}}\,. \end{align}\] Applying Lemma 10 to the second inequality \[\sigma^{\frac{1}{n+1}} \frac{n}{4(n+1)} V(C(S))^{\frac{-(n+2)}{n+1}}|V(C(B_1))-V(C(B_2))|^2 \leq (n+1) (\sigma V(C(S))^n)^{\frac{1}{n+1}}- A(S)\] since \(\max\{V(C(B_1)),V(C(B_2))\}\leq V(C(S))\). As \(V(\{f=t_0\})\leq V(\mathbb{H})=0\), it follows by definition of \(\Omega_1\) and \(\Omega_2\) that \[V(C(S) \Delta B_{t_0}(M)) = V(C(B_1))-V(C(B_2))\,.\] By another application of the BE-inequality, now to \(S\subset M\), we obtain \[\begin{align} V(C(S) \Delta B_{t_0}(M))^2&=(V(C(B_1))-V(C(B_2)))^2 \leq 4(n+1)\frac{(n+1)(\sigma V(C(S))^n)^{\frac{1}{n+1}}- A(S)}{n\sigma^{\frac{1}{n+1}} V(C(S))^{\frac{-(n+2)}{n+1}}} \\ &= \frac{4(n+1)A(S)}{n\sigma^{\frac{1}{n+1}} V(C(S))^{\frac{n}{n+1}}}\delta_{BE} V(C(S))^2 \leq 4\frac{(n+1)^2}{n} \delta_{BE}V(C(S))^2\,. \end{align}\] An application of Lemma 11 finally yields \[A_F(S)^2 \leq 4\tilde{A}_F(S)^2\leq 4 \frac{V(C(S) \Delta B_{t_0}(M))^2}{V(C(S))^2} \leq 16\frac{(n+1)^2}{n} \delta_{BE} \,,\] as claimed. The proof of the optimality of the exponent is part of the next section. 0◻

5 Sharpness of the exponents for the \(L^1\)-stability results↩︎

In this section we show sharpness of the relative behavior between deficit and Fraenkel asymmetry in Theorem 1, Corollary 4, and Corollary 5. To this end, we will give an example of a fixed rotationally symmetric conical Minkowski spacetime \(M\) with a sequence \(S_{\epsilon}\) of smooth achronal hypersurfaces as graphs over \(\mathbb{H}\cap M\) with \(M=\mathbb{R}_+S_{\epsilon}\) such that \(\lim_{\epsilon\to 0}A_F(S_{\epsilon})= 0\), \(i=1,2,3\), and \[\label{eq:optdim} \limsup_{\epsilon\to 0}\frac{\delta_{BE}(S_{\epsilon})}{A_F(S_{\epsilon})^2}<\infty\,, \qquad \limsup_{\epsilon\to 0}\frac{\delta_{CM}(S_{\epsilon})}{A_F(S_{\epsilon})}<\infty\,, \qquad \limsup_{\epsilon\to0}\frac{\delta_{CM}^*(S_{\epsilon})}{A_F(S_{\epsilon})^2}<\infty\,.\tag{17}\] Note that a priori we could choose different sequences \(S_\varepsilon\) of Cauchy hypersurfaces for different inequalities.

Recall the formulas for the volume \(V(C(S))\) in 10 and the area \(A(S)\) in 12 of a rotationally symmetric, non-timelike Lipschitz hypersurface \(S\), in terms of the function \(r\) satisfying \(f(x)=r(d_{\mathbb{H}}(x_0,x))\) for some fixed \(x_0\in \mathbb{H}\).

To prove sharpness of the exponents, we consider a perturbation \(S_\varepsilon\) of a ball \(\bar B_{t^*}(x_0)\) of radius \(t^*>0\) in \(\mathbb{H}\) in the conical Minkowski spacetime \(M=\mathbb{R}_+ \bar B_{t^*}(x_0)\), defined as a graph via \[r_{\varepsilon}(t) \mathrel{\vcenter{:}}= 1 + \varepsilon \varphi(t)\,,\qquad t\in [0,t^*]\,,\] for some smooth, compactly supported function \(\varphi\) with \(\varphi\not\equiv0\) and \(\partial_t^m\varphi(0)=0\) for all \(m\in \mathbb{N}\). The latter condition guarantees smoothness of \(\varphi\) in \(x_0\). Note that for any sufficiently small \(\varepsilon\), the function \(r_\varepsilon\) and the corresponding function \(f_{\varepsilon}\) satisfy the spacelike condition \(|r'_\varepsilon|<r_\varepsilon\) resp. \(|\nabla f_{\varepsilon}|<f_{\varepsilon}\). For such \(\varepsilon\), the smooth hypersurfaces \(S_\varepsilon=S_{f_\varepsilon}\) are achronal by convexity of balls in \(\mathbb{H}\), the discussion in Subsection 2.3, and Lemma 7.

We first expand the geometric quantities that appear in the definitions of the deficit functional \(\delta_{CM}\) given in 3 up to the first nontrivial and non-constant order in \(\varepsilon\). The volume of the cone over the graph parametrized by \(r_\varepsilon\) is given by \[\begin{align} V(C(S_\varepsilon)) &= \frac{1}{n+1} \int_{0}^{t^*} r_{\varepsilon}(t)^{n+1} \,\mathrm d\nu(t) \\&=V(C(S_0))+ \varepsilon \int_{0}^{t^*} \varphi(t) \,\mathrm d\nu(t)+\frac{\varepsilon^2}{2}n \int_{0}^{t^*} \varphi^2(t) \,\mathrm d\nu(t) + \mathcal{O}(\varepsilon^3)\,, \end{align}\] and similarly the corresponding surface volume is given by \[\begin{align} A(S_\epsilon)&=\int_{0}^{t^*} (1+\varepsilon\varphi(t))^{n-1}\sqrt{ 1+2\varepsilon \varphi(t) + \varepsilon^2 \varphi(t)^2 - \varepsilon^2 \varphi'(t)^2}\,\mathrm d\nu(t) \\ & = A(S_0)+\varepsilon n\int_0^{t^*}\varphi(t)\,\mathrm d\nu(t)+\frac{\varepsilon^2}{2}\int_0^{t^*} \left(n(n-1)\varphi^2(t)-(\varphi'(t))^2\right)\,\mathrm d\nu(t) +\mathcal{O}(\varepsilon^3)\,. \end{align}\] Finally, we expand \[\operatorname{dist}(O,S_\varepsilon)= \inf_{t} r_\varepsilon (t)= 1+\varepsilon \inf_{t} \varphi (t)= \operatorname{dist}(O,S_0)+\varepsilon \inf_{t} \varphi (t) \,.\]

Next, we prove that the Fraenkel asymmetry scales linearly in \(\varepsilon\) up to higher order terms. To this end, observe that, for all measure spaces \((X,\mathcal{A}, \iota)\), all \(b\in\mathbb{R}\), and all integrable functions \(h:X\to\mathbb{R}\) with mean \(0\), we have \[\int_X |h(x)+b| \,\mathrm d\iota(x)\geq \int_{\{\operatorname{sgn}(b)h\geq0\}}\operatorname{sgn}(b)h(x)\,\mathrm d\iota(x) = \frac{1}{2}\int_X|h(x)|\,\mathrm d\iota(x)\,.\] Since \(\varphi\) has vanishing mean, we can bound \[\begin{align} V(C(S_\varepsilon) \Delta C(\alpha S_0)) &= \frac{1}{n+1} \int_{0}^{t^*} | r_{\varepsilon}(t)^{n+1}-(\alpha r_0(t))^{n+1}| \,\mathrm d\nu(t)\\ &= \frac{1}{n+1} \int_{0}^{t^*} | \varepsilon(n+1)\varphi(t)+1-\alpha^{n+1}| \,\mathrm d\nu(t)+ \mathcal{O}(\varepsilon^2) \\&\geq \frac{\varepsilon}{2}\int_{0}^{t^*} |\varphi(t)| \,\mathrm d\nu(t) + \mathcal{O}(\varepsilon^2) \end{align}\] independently of \(\alpha\). In particular, we obtain \[A_F(S_\varepsilon) \geq \frac{\varepsilon (n+1)}{2}\fint_{0}^{t^*} |\varphi(t) |\,\mathrm d\nu(t)+\mathcal{O}(\epsilon^2)\,,\] which completes the proof as \(\|\varphi\|_{L^1((0,t^*))}>0\). Here \(\fint\) denotes the integral normalized with respect to the measure \(\mathrm d\nu\).

Note that, since \(0\leq \delta_{BE}(S_\varepsilon)\leq \delta_{CM}^*(S_\varepsilon)\) by Proposition 3, the first bound in 17 is a consequence of the third. Hence, we are left to prove optimality for the CM-inequality and its refinement.

In an asymptotic expansion with a mean-zero \(\varphi\), the first order terms of \(V(C(S_{\epsilon}))\) and \(A(S_{\epsilon})\) in \(\varepsilon\) vanish. For the CM-inequality, we then obtain \[\delta_{CM}(S_\varepsilon)=\frac{(n+1)V(C(S_{\varepsilon}))}{A(S_{\varepsilon})\operatorname{dist}(O,S_{\varepsilon})} -1=- \varepsilon \inf_t\varphi(t)+\frac{\varepsilon^2}{2A(S_0)}\fint_0^{t^*} \left(2n\varphi^2(t)+\varphi'(t)^2\right)\,\mathrm d\nu(t)+\mathcal{O}(\varepsilon^3)\,.\] In particular, we find for the refined version that \[\delta^*_{CM}(S_\varepsilon)=\frac{(n+1)V(C(S_{\varepsilon}))}{A(S_{\varepsilon})\operatorname{dist}(O,S_{\varepsilon})} -1 +\varepsilon\inf_{t}\varphi(t)=\frac{\varepsilon^2}{2A(S_0)}\fint_0^{t^*} \left(2n\varphi^2(t)+\varphi'(t)^2\right)\,\mathrm d\nu(t)+\mathcal{O}(\varepsilon^3)\,.\] By the choice of \(\varphi\), the quotients \[\frac{\delta_{CM}(S_\varepsilon)}{A_F(S_{\varepsilon})}\leq \frac{-2\inf_t\varphi(t)}{(n+1)\fint_{0}^{t^*} |\varphi(t) |\,\mathrm d\nu(t)}+\mathcal{O}(\varepsilon)\] and \[\frac{\delta_{CM}^*(S_\varepsilon)}{A_F(S_{\varepsilon})^2}\leq \frac{2\fint_0^{t^*} \left(2n\varphi^2(t)+(\varphi'(t))^2\right)\,\mathrm d\nu(t)}{A(S_0)(n+1)^2\left(\fint_{0}^{t^*} |\varphi(t) |\,\mathrm d\nu(t)\right)^2}+\mathcal{O}(\varepsilon)\] stay bounded.

6 Complementary stability results↩︎

In this appendix we quantify the functional-analytic proofs from Subsection 3.3 based on Hölder’s and Bernoulli’s inequality in quantitative form. To this end, let \(S\), \(M\), \(f\), \(g\), \(\bar g\), and \(\varphi\) be as used in Subsection 3.3. Assume that \(V(C(S)),V(C(\pi(S)))\in (0,\infty)\) holds similarly to the second proof of Theorem 8.

For Hölder’s inequality, there are various quantitative improvements available; see [59] and [60]. Consequently, we can immediately deduce quantitative versions of the BE-inequality.

Applying [59] if \(p=(n+1)/n\) or \(p= n+1\) and [60] if \(p=2\) yields \[\frac{1}{\beta(p)}\inf_c\left\|\left(\frac{g}{\bar g}\right)^{\frac{1}{p}}-c\,\right\|^{\max\{p,2\}}_{L^{p}(\pi(S))}\leq \frac{\delta_{BM}}{1+\delta_{BM}}\leq \delta_{BM}\,,\label{eq:prelBEstab}\tag{18}\] where \(\beta((n+1)/n)\mathrel{\vcenter{:}}= 4n\), \(\beta(n+1)\mathrel{\vcenter{:}}= (n+1)2^n\), and \(\beta(2)\mathrel{\vcenter{:}}= n+1\). Note that the distances in 18 resemble the Fraenkel-type asymmetry \[\tilde{A}_F(S)=\inf_c\left\|\frac{g}{\bar g}-c\,\right\|_{L^{1}(\pi(S))}\,,\] which is equivalent to the usual Fraenkel asymmetry and defined in Subsection 4.3.

In fact, the elementary inequality \(|a^p-b^p|\geq |a-b|^p\), \(a,b\geq 0\), \(p>1\), tells us that the Fraenkel asymmetry dominates the other distances, so qualitatively speaking, if the Fraenkel asymmetry is small, also the other distances are small. However, the following example shows that the other distances cannot control the Fraenkel asymmetry in a domain-independent way. Hence, we cannot use them to prove our stability estimates with domain-independent constants.

A quantitative version of Bernoulli’s inequality is effectively a second order Taylor expansion. To apply it, we have to require \(\varphi/\bar g\) to be sufficiently small. This gives a stability result in terms of the \(L^2\)-norm. More explicitly, for every \(\tilde{c}\in (0,1/8)\) there is a \(\lambda>0\) such that for all \(\|\varphi\|_{L^\infty(\pi(S))}\leq \lambda \bar g\) we have \[\frac{\tilde{c}}{V(B_1(M))} \inf_c\left\|\frac{g}{\bar g}-c\,\right\|^2_{L^{2}(\pi(S))}\leq \frac{\delta_{BM}}{1+\delta_{BM}}\leq \delta_{BM}\,.\] On the first glance, this stability result is stronger than Theorem 1 as the \(L^2\)-norm is better than the \(L^1\)-norm. However, we have to pay a price, namely the restrictive, pointwise smallness assumption on the perturbation \(\varphi\).

7 An improved stability constant for the refined Cavalletti–Mondino inequality↩︎

In this appendix we provide an alternative argument for the stability of the refined CM-inequality that does not rely on the more involved quantitative BE-inequality but just on the BE-inequality. Moreover, we obtain the new stability constant \[\left(\frac{1}{2^{2+\frac{1}{n+1}}}-\frac{n}{4(n+1)}\right)^{-1}\,,\] which is better than the one provided by Theorem 1 in Corollary 5. Indeed, we observe that \[\frac{1}{n+1}\left(\frac{1}{2^{2+\frac{1}{n+1}}}-\frac{n}{4(n+1)}\right)^{-1}\nearrow \frac{4}{1-\ln(2)}< 16 \swarrow \frac{1}{n+1}\left(16\frac{(n+1)^2}{n}\right)\] as \(n\to \infty\), where \(4(1-\ln(2))^{-1}\approx 13.06\).

References↩︎