Centrally symmetric convex bodies and distributions

To each centrally symmetric convex body is assigned a distribution on the sphere. As applications, geometric formulas and a characterization of zonoids are obtained.     

Centrally symmetric convex bodies and distributions II

In continuation of a previous work we study the generating distributions of centrally symmetric convex bodies and obtain some more geometric formulas and new characterizations of zonoids and generalized zonoids.     

Mean projection and section radii of convex bodies

We introduce new series of mean outer and inner radii, which are defined as the outer (respectively, inner) radius of, either the projection of the convex body onto an i-dimensional subspace, or the i-dimensional section, 1 ≤ i ≤ n, averaged over the Grassmannian manifold, and with respect to the Haar probability measure. We study some properties of these new functionals, establishing inequalities among them, as well as their relation with other measures as the volume or the quermassintegrals.     

Irregularities of point distributions relative to homothetic convex bodies I

The uniformity and irregularities of point distributions can be measured by various kinds of geometric objects. In this paper we prove the existence of point sets that have relatively small irregularities with respect to homothetic copies of a fixed convex body. The results give higher dimensional alternatives to a theorem of Beck.Supported by the Alexander von Humboldt Foundation.     

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     

Inequalities for convex bodies and their polar bodies

The infimum of the quermassintegral product W _i (K)W _i (K*) for i = n – 1 was established by Lutwak. In this paper, the infimum of the dual quermassintegral product ${\widetilde{W}_{n+p}(K)\widetilde{W}_{n+p}(K^*)}$ for any p ≥ 1 is obtained, and some new inequalities about convex bodies and their polar bodies are established.     

Blaschke-decomposable convex bodies

For a given centred convex bodyK of ℝ,n≥3, let be the class of all convex bodies with the same projection body asK. The question whetherK can be expressed as a Blaschke average of two non-homothetic bodies from is considered. Necessary and sufficient conditions onK to be Blaschke decomposable in are given. The paper provides also a characterization of the bodiesK such that the Blaschke indecomposable bodies in are dense in itself.     

A width-diameter inequality for convex bodies

A special case of the Blaschke-Santaló inequality regarding the product of the volumes of polar reciprocal convex bodies is shown to be equivalent to a power-mean inequality involving the diameters and widths of a convex body. This power-mean inequality leads to strengthened versions of various known inequalities.     

Translative integral formulae for convex bodies

Translative versions of the principal kinematic formula for quermassintegrals of convex bodies are studied. The translation integral is shown to be a sum of Crofton type integrals of mixed volumes. As corollaries new integral formulas for mixed volumes are obtained. For smooth centrally symmetric bodies the functionals occurring in the principal translative formula are expressed by measures on Grassmannians which are related to the generating measures of the bodies.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Completions of convex families of convex bodies

The paper discusses the existence of a continuous extension of functions that are defined on subsets of ? ⁿ and whose values are convex bodies in ? ⁿ . This problem arose in convex geometry in connection with the notion, recently introduced in algebraic geometry, of convex Newton-Okounkov bodies.     

Polar reciprocal convex bodies

The minimum of the product of the volume of a symmetric convex bodyK and the volume of the polar reciprocal body ofK relative to the center of symmetry is attained for the cube and then-dimensional crossbody. As a consequence, there is a sharp upper bound in Mahler’s theorem on successive minima in the geometry of numbers. The difficulties involved in the determination of the minimum for unsymmetricK are discussed. Reserch partially supported by NSF Grant GP-27960. An erratum to this article is available at .     

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 reduced convex bodies

A convex bodyK ⊂R ^d is called reduced if for each convex bodyK′ ⊂K,K′ ≠K, the width ofK′ is less than the width ofK. We prove that reduced bodyK is of constant width if (i) the bodyK has a supporting sphere almost everywhere in ∂K. (The radius of the sphere may vary with the point in ∂K; the condition (i) and strict convexity do not imply each other.) Supported by an N.S.E.R.C. Grant of Canada.     

A sharp isoperimetric bound for convex bodies

We consider the problem of lower bounding the Minkowski content of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp for all set sizes, dimensions, and norms. In the case of uniform density a stronger theorem is shown which is also sharp. Supported in part by VIGRE grants at Yale University and the Georgia Institute of Technology.     

The central limit problem for convex bodies

It is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately Gaussian. Several quite broad classes of bodies are shown to satisfy the condition.