Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

Our main result states that the hyperspace of convex compact subsets of a compact convex subset X in a locally convex space is an absolute retract if and only if X is an absolute retract of weight ?ω₁. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube Iω₁ is homeomorphic to Iω₁. An analogous result is also proved for the cone over Iω₁. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.     

Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.     

On the vertex index of convex bodies

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K⊂Rd, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by d² smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. More precisely, we show that for every centrally symmetric convex body K⊂Rd one has

     

On neighborly families of convex bodies

A family of convex bodies in E^d is called neighborly if the intersection of every two of them is (d-1)-dimensional. In the present paper we prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d, d 3, such that every two of them are affinely equivalent (i.e., there is an affine transformation mapping one of them onto another), the bodies have large groups of affine automorphisms, and the volumes of the bodies are prescribed. We also prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d such that the bodies have large groups of symmetries. These two results are answers to a problem of B. Grünbaum (1963). We prove also that there exist arbitrarily large neighborly families of similar convex d-polytopes in E^d with prescribed diameters and with arbitrarily large groups of symmetries of the polytopes.     

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.     

Convex bodies of constant width and constant brightness

In 1926 Nakajima (= Matsumura) showed that any convex body in R³ with constant width, constant brightness, and boundary of class C² is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.     

Asymptotic formulas for the diameter of sections of symmetric convex bodies

Sharpening work of the first two authors, for every proportion λ∈(0,1) we provide exact quantitative relations between global parameters of n-dimensional symmetric convex bodies and the diameter of their random ⌊λn⌋-dimensional sections. Using recent results of Gromov and Vershynin, we obtain an “asymptotic formula” for the diameter of random proportional sections.     

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     

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.     

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.     

A generalized localization theorem and geometric inequalities for convex bodies

In this article, we generalize a localization theorem of Lovász and Simonovits [Random walks in a convex body and an improved volume algorithm, Random Struct. Algorithms 4-4 (1993) 359-412] which is an important tool to prove dimension-free functional inequalities for log-concave measures. In a previous paper [Fradelizi and Guédon, The extreme points of subsets of s-concave probabilities and a geometric localization theorem, Discrete Comput. Geom. 31 (2004) 327-335], we proved that the localization may be deduced from a suitable application of Krein-Milman's theorem to a subset of log-concave probabilities satisfying one linear constraint and from the determination of the extreme points of its convex hull. Here, we generalize this result to more constraints, give some necessary conditions satisfied by such extreme points and explain how it may be understood as a generalized localization theorem. Finally, using this new localization theorem, we solve an open question on the comparison of the volume of sections of non-symmetric convex bodies in Rⁿ by hyperplanes. A surprising feature of the result is that the extremal case in this geometric inequality is reached by an unusual convex set that we manage to identify.     

Densely two-valued metric projections in uniformly convex Banach spaces

For every uniformly convex Banach spaceX with dimX2 there is a residual setU in the Hausdorff metric spaceB(X) of bounded and closed sets inX such that the metric projection generated by a set fromU is two-valued and upper semicontinuous on a dense and everywhere continual subset ofX. For any two closed and separated subsetsM ₁ andM ₂ ofX the points on the equidistant hypersurface which have best approximations both inM ₁ andM ₂ form a dense G set in the induced topology.The author is partially supported by the National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under contract MM 408/94.     

Convex bodies with equiframed two-dimensional sections

We show that a centred, convex body in (d ≥ 3) all of whose two-dimensional sections through the origin are equiframed is an ellipsoid. Received: 20 July 2005 Revised: 16 May 2006     

A note on measures of nonconvexity

We define the Hausdorff measure of nonconvexity β(C) of a nonempty bounded subset C of a Banach space X as the Hausdorff distance of C to the family of all the nonempty convex bounded subsets of X. We compare the measure β with the Eisenfeld-Lakshmikantham measure of nonconvexity α and prove that the two measures are equivalent (β≤α≤2β), but in general they are different.     

The number of weakly compact convex subsets of the Hilbert space

We prove that for κ an uncountable cardinal, there exist κ² many nonhomeomorphic weakly compact convex subsets of weight κ in the Hilbert space ?₂(κ).     

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.     

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.     

Affine images of compact convex sets and maximal measures

Let be an affine continuous mapping of a compact convex set X onto a compact convex set Y. We show that the induced mapping φ_? need not map maximal measures on X to maximal measures on Y even in case φ maps extreme points of X to extreme points of Y. This disproves Théorème 6 of [S. Teleman, Sur les mesures maximales, C. R. Acad. Sci. Paris Sér. I Math. 318 (6) (1994) 525-528]. We prove the statement of Théorème 6 under an additional assumption that extY is Lindelöf or Y is a simplex. We also show that under either of these two conditions injectivity of φ on extX implies injectivity of φ_? on maximal measures. A couple of examples illustrate the results.