On the extremal distance between two convex bodies (Q636023)

Let \( d(K,L) = \inf\{|\lambda| : \lambda \in {\mathbb R}, K - a \subset T(L - b) \subset \lambda (K - a) \}\) be the modified Banach-Mazur distance between two convex bodies \(K\) and \( L \) in \({\mathbb R^n}\). Here the infimum is taken over all \(a, b \in {\mathbb R^n}\) and all \( T \in GL_n \). The authors study a natural conjecture: If \(K, L \subset {\mathbb R^n}\) are convex bodies, and \(d(K,L) = n \), then one of these bodies is a simplex. The main result of the paper asserts that if \(K\) is a convex body, \(L\) a strictly convex or a smooth convex body in \({\mathbb R^n}\), and \(d(K,L) = n \), then \(K\) is a simplex. The authors also show that the strict convexity (smoothness) condition is essential here.