| Résume | In 1970, Schneider introduced the mth order difference body of a convex body, and conjectured its volume is minimized by ellipsoids. Even now, 55 years later, this question remains open for m greater than or equal to 2 in dimensions 3 and higher. In this talk, we discuss the history of this problem and related results. We then solve the dual problem of maximizing the volume of the polar of the mth order difference body. As a special case, one recovers the celebrated Blaschke-Santaló inequality. As an application, we establish Schneider's conjectured inequality up to an absolute constant. |
