Minkowski Additive Operators Under Volume Constraints

Abstract

We investigate Minkowski additive, continuous, and translation invariant operators Φ : K n → K n defined on the family of convex bodies such that the volume of the image Φ(K) is bounded from above and below by multiples of the volume of the convex body K, uniformly in K. We obtain a representation result for an infinite subcone contained in the cone formed by this type of operators. Under the additional assumption of monotonicity or SO(n)-equivariance, we obtain new characterization results for the difference body operator.

A Characterization of the Polarity Mapping for Convex Bodies

“…For more results on the characterization of duality and lattice endomorphism of the class of convex bodies and of convex sets, we can refer to Refs. [25][26][27][28][29].…”

Blaschke-Santaló inequalities for Minkowski and Asplund endomorphisms

“…This theorem was a byproduct of a more general, systematic study of Minkowski additive operators on K n , initiated about 50 years ago by Schneider [49][50][51]. Since then, and up to now, the main focus thereby has been on maps that also commute with SO(n) transforms (see [1,21,31,[53][54][55]). As such maps are automatically compatible with translations (see, e.g., [31,Section 2.3]), they are often assumed w.l.o.g.…”

Radial Minkowski additive operators

“…In the 1930s, Blaschke started a systematic investigation, and then Hadwiger obtained the famous Hadwiger's characterization theorem. The Hadwiger's characterization theorem provides the connection between rigid motion invariant set functions and symmetric polynomials (see [4]) for further results and generalizations, see [1], [2], [3], [5], [20], [29]. The following Minkowski endomorphism was introduced by Schneider in [21].…”