A new affine invariant for polytopes and Schneider's projection problem

New affine invariant functionals for convex polytopes are introduced. Some sharp affine isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new affine isoperimetric inequalities are extensions of Ball's reverse isoperimetric inequalities.

     

A reciprocity principle for constrained isoperimetric problems and existence of isoperimetric subregions in convex sets

It is a well known fact that in \(\mathbb {R} ^n\) a subset of minimal perimeter L among all sets of a given volume is also a set of maximal volume among all sets of the same perimeter L. This is called the reciprocity principle for isoperimetric problems. The aim of this note is to prove this relation in the case where the class of admissible sets is restricted to the subsets of some subregion \(G\subsetneq \mathbb {R} ^n\). Furthermore, we give a characterization of those (unbounded) convex subsets of \(\mathbb {R} ^2\) in which the isoperimetric problem has a solution. The perimeter that we consider is the one relative to \(\mathbb {R} ^n\).     

A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones

The relative isoperimetric inequality inside an open, convex cone $\mathcal{C}$ states that, at fixed volume, $B_{r} \cap\mathcal{C}$ minimizes the perimeter inside $\mathcal{C}$ . Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov’s proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside $\mathcal{C}$ . Our proof follows the line of reasoning in Figalli et al.: Invent. Math. 182:167–211 (2010), though several new ideas are needed in order to deal with the lack of translation invariance in our problem.     

Isoperimetric estimates in products

In a product M ₁ × M ₂ of Riemannian manifolds, the least perimeter required to enclose given volume among general regions is at least 1/√ 2 times that among regions of product form, assuming that the isoperimetric profiles of M ₁ and M ₂ are concave. This result sharpens earlier work of Grigor'yan, generalizes results of Bollobás and Leader and of Barthe, and yields a lower bound on the Cheeger isoperimetric constant of a product.     

Total variation and Cheeger sets in Gauss space

The aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the Gauss space, in both the finite and infinite dimensional case. We first study the finite dimensional case, proving the existence of a maximal Cheeger set which is convex inside any bounded convex set. We also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set. Then we extend these results in the context of the abstract Wiener space, and for that we study the total variation denoising problem in this context.     

A Brascamp-Lieb-Luttinger-type inequality and applications to symmetric stable processes

We derive an inequality for multiple integrals from which we conclude various generalized isoperimetric inequalities for Brownian motion and symmetric stable processes in convex domains of fixed inradius. Our multiple integral inequality is a replacement for the classical inequality of H. J. Brascamp, E. H. Lieb and J. M. Luttinger, where instead of fixing the volume of the domain one fixes its inradius.

     

On the wellposedness of constitutive laws involving dissipation potentials

We consider a material with memory whose constitutive law is formulated in terms of internal state variables using convex potentials for the free energy and the dissipation. Given the stress at a material point depending on time, existence of a strain and a set of inner variables satisfying the constitutive law is proved. We require strong coercivity assumptions on the potentials, but none of the potentials need be quadratic.

As a technical tool we generalize the notion of an Orlicz space to a cone ``normed' by a convex functional which is not necessarily balanced. Duality and reflexivity in such cones are investigated.

     

Minimizers for nonlocal perimeters of Minkowski type

We study a nonlocal perimeter functional inspired by the Minkowski content, whose main feature is that it interpolates between the classical perimeter and the volume functional. This nonlocal functionals arise in concrete applications, since the nonlocal character of the problems and the different behaviors of the energy at different scales allow the preservation of details and irregularities of the image in the process of removing white noises, thus improving the quality of the image without losing relevant features. In this paper, we provide a series of results concerning existence, rigidity and classification of minimizers, compactness results, isoperimetric inequalities, Poincaré–Wirtinger inequalities and density estimates. Furthermore, we provide the construction of planelike minimizers for this generalized perimeter under a small and periodic volume perturbation.     

锥体积泛函与仿射不等式(英文)

本文研究了凸多胞形的锥体积泛函.利用投影体以及Lutwak、杨和张最近所建立的仿射等周不等式,得到了刻划平行四边形特征的一个崭新不等式和用锥体积泛函以及投影体的体积所表达的关于配极体体积的严格下界.     

A Discrete Isoperimetric Inequality on Lattices

We establish an isoperimetric inequality with constraint by \(n\) -dimensional lattices. We prove that, among all sets which consist of lattice translations of a given rectangular parallelepiped, a cube is the best shape to minimize the ratio involving its perimeter and volume as long as the cube is realizable by the lattice. For its proof a solvability of finite difference Poisson–Neumann problems is verified. Our approach to the isoperimetric inequality is based on the technique used in a proof of the Aleksandrov–Bakelman–Pucci maximum principle, which was originally proposed by Cabré (Butll Soc Catalana Mat 15:7–27, 2000) to prove the classical isoperimetric inequality.     

A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains

We show that as the ratio between the first Dirichlet eigenvalues of a convex domain and of the ball with the same volume becomes large, the same must happen to the corresponding ratio of isoperimetric constants. The proof is based on the generalization to arbitrary dimensions of Pólya and Szegö's upper bound for the first eigenvalue of the Dirichlet Laplacian on planar star-shaped domains which depends on the support function of the domain.

As a by-product, we also obtain a sharp upper bound for the spectral gap of convex domains.

     

Sharp $$N^{3/4}$$ Law for the Minimizers of the Edge-Isoperimetric Problem on the Triangular Lattice

We investigate the edge-isoperimetric problem (EIP) for sets of n points in the triangular lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. By introducing a suitable notion of perimeter and area, EIP minimizers are characterized as extremizers of an isoperimetric inequality: they attain maximal area and minimal perimeter among connected configurations. The maximal area and minimal perimeter are explicitly quantified in terms of n. In view of this isoperimetric characterizations, EIP minimizers \(M_n\) are seen to be given by hexagonal configurations with some extra points at their boundary. By a careful computation of the cardinality of these extra points, minimizers \(M_n\) are estimated to deviate from such hexagonal configurations by at most \(K_t\, n^{3/4}+\mathrm{o}(n^{3/4})\) points. The constant \(K_t\) is explicitly determined and shown to be sharp.     

The Minkowski problem for polytopes

The traditional solution to the Minkowski problem for polytopes involves two steps. First, the existence of a polytope satisfying given boundary data is demonstrated. In the second step, the uniqueness of that polytope (up to translation) is then shown to follow from the equality conditions of Minkowski's inequality, a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context. In this article we adapt the classical argument to prove both the existence theorem of Minkowski and his mixed volume inequality simultaneously, thereby providing a new proof of Minkowski's inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry.     

Positivity of the Shape Hessian and Instability of some Equilibrium Shapes

We study the positivity of the second shape derivative around an equilibrium for a 2-dimensional functional involving the perimeter of the shape and its the Dirichlet energy under volume constraint. We prove that, generally, convex equilibria lead to strictly positive second derivatives. We also exhibit some examples where strict positivity of the second order derivative holds at an equilibrium while existence of a minimum does not.     

Extremal Polygons with Minimal Perimeter

In this paper we will investigate an isoperimetric type problem in lattices. If K is a bounded O-symmetric (centrally symmetric with respect to the origin) convex body in En of volume v(K) = 2n det L which does not contain non-zero lattice points in its interior, we say that K is extremal with respect to the given lattice L. There are two variations of the isoperimetric problem for this class of polyhedra. The first one is: Which bodies have minimal surface area in the class of extremal bodies for a fixed n-dimensional lattice? And the second one is: Which bodies have minimal surface area in the class of extremal bodies with volume 1 of dimension n? We characterize the solutions of these two problems in the plane. There is a consequence of these results, the solutions of the above problems in the plane give the solution of the lattice-like covering problem: Determine those centrally symmetric convex bodies whose translated copies (with respect to a fixed lattice L) cover the space and have minimal surface area.

     

Approximation of Convex Bodies and a Momentum Lemma for Power Diagrams

 The volume of the symmetric difference of a smooth convex body in and its best approximating polytope with n vertices is asymptotically a constant multiple of . We determine this constant and the similarly defined constant for approximation with a given number of facets by solving two isoperimetric problems for planar tilings. Received 15 May 1997; in revised form 14 August 1997     

An existence theorem of harmonic functions with polynomial growth

We prove the existence of nonconstant harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature, Euclidean volume growth and unique tangent cone at infinity.

     

Some comments on floating and centroid bodies in the plane

Consider a long, convex, homogenous cylinder with horizontal axis and with a planar convex body K as transversal section. Suppose the cylinder is immersed in water and let \(K_w\) be the wet part of K. In this paper we study some properties of the locus of the centroid of \(K_w\) and prove an analogous result to Klamkin–Flanders’ theorem when the locus is a circle. We also study properties of bodies floating at equilibrium when either the origin or the centroid of the body is pinned at the water line. In some sense this is the floating body problem for a density varying continuously. Finally, in the last section we give an isoperimetric type inequality for the perimeter of the centroid body (defined by C. M. Petty in Pacific J Math 11:1535–1547, 1961) of convex bodies in the plane.     

A variant of the isoperimetric problem,PartⅠ

This paper deals with the following isoperimetric problem in the plane:Among all regions with prescribed perimeter and covering a given line segment,what is the region that has the greatest area?