Minkowski Endomorphisms

Abstract

(Duke Math J 162:1895-1922, 2013.

Minkowski Additive Operators Under Volume Constraints

“…The claim will follow if for every ǫ > 0, there exists i 0 ∈ I k such that d(E i , E) < ǫ for every i > i 0 , i ∈ I k . Using expression (13) and (12), we obtain easily that this condition is satisfied. Indeed, since the sequence (v i j ) i∈I k tends to v j for every 1 ≤ j ≤ k, the distance, i.e., the angle, between span{v i j } and span{v j } also tends to zero.…”

“…Since their introduction, Minkowski endomorphisms and their extensions have been widely studied and different representation results have been obtained, usually under further geometric assumptions (see, for instance, [27,48,49,52,53,55,56]). We also point out the recent work by Dorrek [13], who carried out a deep systematic study of Minkowski endomorphisms closing some of the most important conjectures related to them within convex geometry.…”

“…Indeed, since the sequence (v i j ) i∈I k tends to v j for every 1 ≤ j ≤ k, the distance, i.e., the angle, between span{v i j } and span{v j } also tends to zero. Since in (13) we are considering a minimum among all distances between 1-dimensional vector spaces, i.e., lines where one of them is fixed, the claim follows.…”

“…MEnd: Minkowski additive, continuous, and translation invariant operators, which are additionally rotation equivariant, i.e., they commute with rotations of R n . Following the current notation (see for instance [50] or [13]), we will refer to these operators as Minkowski endomorphisms and denote their class by MEnd. Minkowski endomorphisms were first introduced in 1974 by Schneider [47].…”

“…Weakly monotonic endomorphisms were characterized by Kiderlen [27], who proved that they are given by certain measures on the sphere. Before the recent results by Dorrek [13] came to light, all known Minkowski endomorphism were weakly monotonic. On the other hand, Theorem 1.2 proves, in particular, that all Minkowski endomorphisms satisfying (VC) are weakly monotonic.…”