Uniqueness of the Cheeger set of a convex body

We prove that if C⊂R^N$C\subset {\mathbb{R}}^N$ is an open bounded convex set, then there is only one Cheeger set inside C$C$ and it is convex. A Cheeger set of C$C$ is a set which minimizes the ratio perimeter over volume among all subsets of C$C$.     

The cross-section body, plane sections of convex bodies and approximation of convex bodies, I

For a convex body K ^d we investigate three associated bodies, its intersection body IK (for 0int K), cross-section body CK, and projection body IIK, which satisfy IKCKIIK. Conversely we prove CKconst₁(d)I(K–x) for some xint K, and IIKconst₂ (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L ^d a convex body, we take n random segments in L, and consider their Minkowski average D. We prove that, for V(L) fixed, the supremum of V(D) (with also nN arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M ^d a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1p<) also are minimized, for V(L) fixed, by the ellipsoids. For k=2, the supremum (nN arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.Research (partially) supported by Hungarian National Foundation for Scientific Research, Grant No. 41.     

How to hold a convex body?

Convex bodies are often used for mathematical tests. They occasionally try to escape. Can the testing mathematician hold them still by using a circle? Rarely not.     

Constructing infinite families of regular incidence (quasi-)polytopes

Given a regular incidence (quasi-)polytopeP of type {a ₁,a ₂, ...,a _n–1} and a function on its directed edges satisfying certain conditions, we construct for everym 2 a regular incidence (quasi-)polytope of type {ma ₁,a ₂, ...,a _n–1} with the same vertex figure asP.     

The image of a closed convex set under a Fredholm operator

The purpose of this article is two-fold. In the first place, we prove that a set is the image of a non empty closed convex subset of a real Banach space under an onto Fredholm operator of positive index if and only if it can be written as the union of {_Dn:n∈N}$\{D_n:n\in \mathbb{N}\}$, a non-decreasing family of non empty, closed, convex and bounded sets such that _Dn+_Dn+2⊆2_Dn+1$D_n+D_{n+2}\subseteq 2D_{n+1}$ for every n∈N$n\in \mathbb{N}$.     

A lower bound on the number of sharp shadow-boundaries of convex polytopes

We give the lower bound on the number of sharp shadow-boundaries of convexd-polytopes (or unbounded convex polytopal sets) withn facets. The polytopes (sets) attaining these bounds are characterized. Additionally, our results will be transferred to the dual theory.The research work of the first author was (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1812.     

Puzzles and polytope isomorphisms

We show that an isomorphism between the graphs of two simple polytopes of arbitrary dimension can always be extended to an isomorphism between the polytopes themselves. It has been convenient to study the dual situation, involving what we like to call the puzzle of a simplicial polytope.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

On the smallest disk containing a planar reduced convex body

A convex body R of Euclidean space E ^d is said to be reduced if every convex body $ P \subset R $ different from R has thickness smaller than the thickness $ \Delta(R) $ of R. We prove that every planar reduced body R is contained in a disk of radius $ {1\over 2}\sqrt 2 \cdot \Delta(R) $. For $ d \geq 3 $, an analogous property is not true because we can construct reduced bodies of thickness 1 and of arbitrarily large diameter.     

On the Sobolev distance of convex bodies

Summary LetK ^d denote the cone of all convex bodies in the Euclidean spaceK ^d . The mappingK h _K of each bodyK K ^d onto its support function induces a metric _w onK ^d by" _w (K, L)h _L –h _{K w} where _w is the Sobolev I-norm on the unit sphere . We call _w (K, L) the Sobolev distance ofK andL. The goal of our paper is to develop some fundamental properties of the Sobolev distance.     

Sections and projections of homothetic convex bodies

For a pair of convex bodies K₁ and K₂ in Euclidean space , n ≥ 3, possibly unbounded, we show that K₁ is a translate of K₂ if either of the following conditions holds: (i) the orthogonal projections of K₁ on 2-dimensional planes are translates of the respective orthogonal projections of K₂, (ii) there are points p₁ ∈K₁ and p₂ ∈K₂ such that for every pair of parallel 2-dimensional planesL₁and L₂ through p₁ and p₂, respectively, the section K₁ ∩ L₁is a translate of K₂ ∩ L₂.     

On the relative distances of seven points in a plane convex body

Let C be a convex body in the Euclidean plane. The relative distance of points p and q is twice the Euclidean distance of p and q divided by the Euclidean length of a longest chord in C with the direction, say, from p to q. We prove that, among any seven points of a plane convex body, there are two points at relative distance at most one, and one cannot be replaced by a smaller value. We apply our result to determine the diameter of point sets in normed planes. Zsolt Lángi: Partially supported by the Hung. Nat. Sci. Found. (OTKA), grant no. T043556 and T037752 and by the Alberta Ingenuity Fund.     

Reconstructing dimensions of intersections of convex sets

Denoting by dimA the dimension of the affine hull of the setA, we prove that if {K _i:i T} and {K _i ^j :i T} are two finite families of convex sets inR ⁿ and if dim {K _i :i S} = dim {K _i ^j :i S}for eachS T such that|S| n + 1 then dim {K _i :i T} = dim {K _i : {i T}}.     

On closed convex sets without boundary rays and asymptotes

We study in finite-dimensional spaces the class of closed convex sets without boundary rays and asymptotes, denoted by and introduced by D. Gale and V. Klee. These sets, not necessarily bounded, enjoy many properties satisfied by compacts sets. New properties of this class are given and convergence analysis of this class is investigated. We also introduce the class of closed convex proper functions which have an epigraph in and we give some properties of these functions.     

Minimal and reduced pairs of convex bodies

We give a new proof for the existence and uniqueness (up to translation) of plane minimal pairs of convex bodies in a given equivalence class of the Hörmander-R»dström lattice, as well as a complete characterization of plane minimal pairs using surface area measures. Moreover, we introduce the so-called reduced pairs, which are special minimal pairs. For the plane case, we characterize reduced pairs as those pairs of convex bodies whose surface area measures are mutually singular. For higher dimensions, we give two sufficient conditions for the minimality of a pair of convex polytopes, as well as a necessary and sufficient criterion for a pair of convex polytopes to be reduced. We conclude by showing that a typical pair of convex bodies, in the sense of Baire category, is reduced, and hence the unique minimal pair in its equivalence class.     

The determination of convex bodies from derivatives of section functions

It is known that non-symmetric convex bodies generally cannot be characterized by the volumes of hyperplane sections through one interior point. Falconer and Gardner, however, independently proved that volumes of hyperplane sections through two different interior points determine the body uniquely. We prove that if −1 < q < n − 1 is not an integer, then the derivatives of the order q at zero of parallel section functions at one interior point completely characterize convex bodies in . If 0 ≤ q < n − 1 is an integer then one needs the derivatives of order q at two different interior points (except for the case where q = n − 2, q odd), generalizing the results of Falconer and Gardner. The first named author was partially supported by the NSF grant DMS 0455696. Received: 31 January 2006     

Duality of convex bodies

The paper presents a category theoretical approach to the notion of duality of convex bodies. Using results of I. Barany (Acta Sci. Math. (Szeged)52 (1988), 93–100), we define and study metric duality , whose advantage is that congruent convex bodies have congruent duals.Dedicated to Professor Helmut Salzmann on the occasion of his 65th birthday     

Positions of convex bodies associated to extremal problems and isotropic measures

We show that there are close relations between extremal problems in dual Brunn-Minkowski theory and isotropic-type properties for some Borel measures on the sphere. The methods we use allow us to obtain similar results in the context of Firey-Brunn-Minkowski theory. We also study reverse inequalities for dual mixed volumes which are related with classical positions, such as ?-position or isotropic position.     

On the volume of convex hulls of sets on spheres

We prove that for a measurable subset of S ^n–1 with fixed Haar measure, the volume of its convex hull is minimized for a cap (i.e. a ball with respect to the geodesic measure). We solve a similar problem for symmetric sets and n=2, 3. As a consequence, we deduce a result concerning Gaussian measures of dilatations of convex, symmetric sets in R ² and R ³.Partially supported by KBN (Poland), Grant No. 2 1094 91 01.     

On low-dimensional faces that high-dimensional polytopes must have

We prove that every five-dimensional polytope has a two-dimensional face which is a triangle or a quadrilateral. We state and discuss the following conjecture: For every integerk1 there is an integer f(k) such that everyd-polytope,df(k), has ak-dimensional face which is either a simplex or combinatorially isomorphic to thek-dimensional cube.We give some related results concerning facet-forming polytopes and tilings. For example, sharpening a result of Schulte [25] we prove that there is no face to face tiling of ⁵ with crosspolytopes.Supported in part by a BSF Grant and by I.H.E.S, Bures-Sur-Yvette.