Self-dual sets up to a translation: a negative answer to Milman's question

Abstract

Let \(K,T\subset\mathbb R^n\) be convex bodies with the origin in their interiors. If \(K=T^\circ\), then the identity \( K+T=K^\circ+T^\circ \) holds trivially. V.~Milman asked whether these trivial solutions are the only ones, that is, whether \( K+T=K^\circ+T^\circ \) forces \(K=T^\circ\). We show that the answer is positive for polytopes but negative for general convex bodies.