Dual mean Minkowski measures of symmetry for convex bodies

Abstract

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. Keywordsgeometric invariant, measure of symmetry, dual measure of symmetry, simplex, affine diameter MSC(2010) 52A20, 46B99, 52A38, 52A39, 52A40Citation: Guo Q, Toth G. Dual mean Minkowski measures of symmetry for convex bodies.