The only convex body with extremal distance from the ball is the simplex (Q1802792)

For convex bodies \(K_ 1\) and \(K_ 2\) in \(\mathbb{R}^ n\) one can define an extension of the Banach-Mazur distance by \[ d(K_ 1,K_ 2):=\inf\{\alpha| \exists x,y\in\mathbb{R}^ n,\;T\in GL(n):y+K_ 1\subset T(x+K_ 2)\subset\alpha(y+K_ 1)\}. \] If \(D\) is a ball, it is known that \(d(K,D)\leq n\) for every convex body \(K\), with equality if \(K\) is a simplex. In this paper it is proved that equality holds only if \(K\) is a simplex.