On asphericity of convex bodies in linear normed spaces.

In 1960, Dvoretzky proved that in any infinite dimensional Banach space and for any [Formula: see text] there exists a subspace of of arbitrary large dimension -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 in a linear normed space, and the center and the radius of the largest ball contained in it provided that 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.