Curvature measures of convex bodies

Summary The curvature measures, introduced by Federer for the sets of positive reach, are investigated in the special case of convex bodies. This restriction yields additional results. Among them are:(5.1), an integral-geometric interpretation of the curvature measure of order m, showing that it measures, in a certain sense, the affine subspaces of codimension m+1 which touch the convex body;(6.1), an axiomatic characterization of the (linear combinations of) curvature measures similar to Hadwiger's characterization of the quermassintegrals of convex bodies;(8.1), the determination of the support of the curvature measure of order m, which turns out to be the closure of the m-skeleton of the convex body. Moreover we give, for the case of convex bodies, a new and comparatively short proof of an integral-geometric kinematic formula for curvature measures. Entrata in Redazione il 14 dicembre 1976.     

Translation-invariant measures of planes intersecting convex bodies

An expression is found for a translation-invariant measure on the set of planes intersecting a convex body.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 146–157, 1987.     

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.     

New affine measures of symmetry for convex bodies

Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with “enough symmetries”.To define these new measures of symmetry, we use affine covariant points. We give examples of convex bodies whose affine covariant points are “far apart”. In particular, we give an example of a convex body whose centroid and Santaló point are “far apart”.     

Reconstruction of convex bodies from surface tensors

We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank s for some natural number s. When only measurements subject to noise of surface tensors are available for reconstruction, we recommend to use certain values of the surface tensors, namely harmonic intrinsic volumes instead of the surface tensors evaluated at the standard basis. The second algorithm we present is based on harmonic intrinsic volumes and allows for noisy measurements. From a generalized version of Wirtinger's inequality, we derive stability results that are utilized to ensure consistency of both reconstruction procedures. Consistency of the reconstruction procedure based on measurements subject to noise is established under certain assumptions on the noise variables.     

Convex bodies and norms associated to convex measures

Isotropy-like properties are considered for finite measures with heavy tails. As a basic tool, we extend K. Ball’s relationship between convex bodies and finite logarithmically concave measures to a larger class of distributions, satisfying convexity conditions of the Brunn–Minkowski type.     

Lattice coverings and Gaussian measures of n-dimensional convex bodies

Let be the euclidean norm on R ⁿ and let γ _n be the (standard) Gaussian measure on R ⁿ with density . Let be defined by and let L be a lattice in R ⁿ generated by vectors of norm ≤ϑ. Then, for any closed convex set V in R ⁿ with , we have (equivalently, for any . The above statement can also be viewed as a ``nonsymmetric' version of the Minkowski theorem. Received March 6, 1995, and in revised form January 26, 1996.     

Balancing vectors and Gaussian measures of n-dimensional convex bodies

Let ‖·‖ be the Euclidean norm on R ⁿ and γ_n the (standard) Gaussian measure on R ⁿ with density (2π)^−n/2e. It is proved that there is a numerical constant c>0 with the following property: if K is an arbitrary convex body in R ⁿ with γ_n(K)≥1/2, then to each sequence u₁,…,u_m∈ R ⁿ with ‖u₁‖,…,‖u_m‖≤c there correspond signs ε₁,…,ε_m=±1 such that ∑^m_i=1ε_iu_i∈K. This improves the well-known result obtained by Spencer [Trans. Amer. Math. Soc. 289 , 679–705 (1985)] for the n-dimensional cube. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 351–360, 1998     

Dual mean Minkowski measures of symmetry for convex bodies

We introduce and study a sequence of geometric invariants for convex bodies in finite-dimensional spaces, which is in a sense dual to the sequence of mean Minkowski measures of symmetry proposed by the second author. It turns out that the sequence introduced in this paper shares many nice properties with the sequence of mean Minkowski measures, such as the sub-arithmeticity and the upper-additivity. More meaningfully, it is shown that this new sequence of geometric invariants, in contrast to the sequence of mean Minkowski measures which provides information on the shapes of lower dimensional sections of a convex body, provides information on the shapes of orthogonal projections of a convex body. The relations of these new invariants to the well-known Minkowski measure of asymmetry and their further applications are discussed as well.     

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.     

On convex bodies of constant width

We present an alternative proof of the following fact: the hyperspace of compact closed subsets of constant width in Rn is a contractible Hilbert cube manifold. The proof also works for certain subspaces of compact convex sets of constant width as well as for the pairs of compact convex sets of constant relative width. Besides, it is proved that the projection map of compact closed subsets of constant width is not 0-soft in the sense of Shchepin, in particular, is not open.