Conic volume ratio of the packing cone associate to a convex body

摘要

Given a convex body ${Ksubsetmathbb{R}^n}$ (n ≥ 1) which contains o in its interior and ${{bf u} in S^{n-1}}$ , we introduce conic volume ratio r(K, u) of K in the direction of u by $$r(K, {bf u})=frac{vol(cone(K,{bf u})cap B_2^n)}{vol(B_2^n)},$$ where cone(K, u) is the packing cone of K in the direction of u. We prove that if K is an o-symmetric convex body in ${mathbb{R}^n}$ and r(K, u) is a constant function of u, then K must be a Euclidean ball.