Premium
Shadow Systems and Volumes of Polar Convex Bodies
Author(s) -
Meyer Mathieu,
Reisner Shlomo
Publication year - 2006
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/s0025579300000061
Subject(s) - mathematics , regular polygon , reciprocal , shadow (psychology) , polytope , affine transformation , combinatorics , volume (thermodynamics) , convex body , mixed volume , function (biology) , point (geometry) , pure mathematics , convex polytope , polar , mathematical analysis , geometry , convex analysis , convex hull , convex optimization , physics , psychology , astronomy , psychotherapist , philosophy , linguistics , quantum mechanics , evolutionary biology , biology
It is proved that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies K t , is a convex function of t , thus extending to the non‐symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santaló inequality for polytopes in ℝ d that have at most d + 3 vertices.