On asphericity of convex bodies in linear normed spaces

Abstract

Abstract In 1960, Dvoretzky proved that in any infinite dimensional Banach space X and for any ϵ>0 $\epsilon> 0$ there exists a subspace L of X of arbitrary large dimension ϵ-iometric to Euclidean space. A main tool in proving this deep result was some results concerning asphericity of convex bodies. In this work, we introduce a simple technique and rigorous formulas to facilitate calculating the asphericity for each set that has a nonempty boundary set with respect to the flat space generated by it. We also give a formula to determine the center and the radius of the smallest ball containing a nonempty nonsingleton set K in a linear normed space, and the center and the radius of the largest ball contained in it provided that K has a nonempty boundary set with respect to the flat space generated by it. As an application we give lower and upper estimations for the asphericity of infinite and finite cross products of these sets in certain spaces, respectively.

Keywords

• Dvoretzky theorem
• Cross product of convex bodies
• Asphericity
• Convex set