Premium
Rotation Invariant Minkowski Classes of Convex Bodies
Author(s) -
Schneider Rolf,
Schuster Franz E.
Publication year - 2007
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300000152
Subject(s) - mathematics , minkowski space , convex body , support function , regular polygon , minkowski addition , mixed volume , invariant (physics) , pure mathematics , convex set , euclidean space , mathematical analysis , combinatorics , convex hull , geometry , convex optimization , mathematical physics
A Minkowski class is a closed subset of the space of convex bodies in Euclidean space ℝ n which is closed under Minkowski addition and non‐negative dilatations. A convex body in ℝ n is universal if the expansion of its support function in spherical harmonics contains non‐zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T 1 , T 2 such that M + T 1 = T 2 , and T 1 , T 2 belong to the rotation invariant Minkowski class generated by K . It is shown that every convex body K which is not centrally symmetric has a linear image, arbitrarily close to K , which is universal. A modified version of the result holds for centrally symmetric convex bodies. In this way, a result of S. Alesker is strengthened, and at the same time given a more elementary proof.