Courtade's conjecture on volumes of Minkowski sums with the ball

Courtade conjectured that for convex bodies $B,C\subset\mathbb R^n$, $$(|B||C|)^{1/n} + (|B_2^n||B_2^n+B+C|)^{1/n} \le |B_2^n+B|^{1/n}|B_2^n+C|^{1/n}.$$ The paper proves that this inequality is false in every dimension $n\ge3$. In dimension $3$, it gives an explicit geometric counterexample using two orthogonal double bodies of revolution. It then strengthens this by constructing zonoid counterexamples in every dimension $n\ge3$, including a six-generator zonotope in dimension $3$ and smooth perturbative constructions in higher dimensions.

Courtade conjectured that for convex bodies $B,C\subset\mathbb R^n$, $$(|B||C|)^{1/n} + (|B_2^n||B_2^n+B+C|)^{1/n} \le |B_2^n+B|^{1/n}|B_2^n+C|^{1/n}.$$ The paper proves that this inequality is false in every dimension $n\ge3$. In dimension $3$, it gives an explicit geometric counterexample using two orthogonal double bodies of revolution. It then strengthens this by constructing zonoid counterexamples in every dimension $n\ge3$, including a six-generator zonotope in dimension $3$ and smooth perturbative constructions in higher dimensions.