On covering functionals of convex bodies

摘要

In the present paper we investigate close connections between the combinatorial geometry of convex bodies and Banach space theory. Inspired by the still unsettled covering problem of Hadwiger (asking for the least number of smaller homothets of a convex body K sufficient to cover K), we derive new results on covering functionals of convex bodies which are closely related to this famous problem. In addition, we show that for the subcase that K is centrally symmetric (and thus can be interpreted as the unit ball of a normed space), these investigations yield new results involving moduli of convexity, James and Schaffer constants and other notions from Banach space theory. (C) 2016 Elsevier Inc. All rights reserved.