The log-Brunn–Minkowski conjecture

Let $K$ and $L$ be convex bodies in $\mathbb{R}^n$. For $\lambda\in(0,1)$, their geometric mean is defined, for origin-symmetric bodies, by

$$\lambda\cdot K+_0(1-\lambda)\cdot L:=\{x\in\mathbb{R}^n:\langle x,u\rangle\leq h_K^{\lambda}(u)h_L^{1-\lambda}(u)\ \forall u\in S^{n-1}\}.$$

Here $V$ denotes volume, $\operatorname{int}$ denotes interior, and $o$ is the origin. Log-Brunn–Minkowski conjecture. For any pair $K$ and $L$ of convex bodies in $\mathbb{R}^n$, there exist $z_K\in\operatorname{int}K$ and $z_L\in\operatorname{int}L$ such that, for every $\lambda\in(0,1)$,

$$V\bigl((1-\lambda)\cdot(K-z_K)+_0\lambda\cdot(L-z_L)\bigr)\geq V(K)^{1-\lambda}V(L)^\lambda.$$

If $K$ and $L$ are origin symmetric, then $z_K=z_L=o$. Moreover, equality holds if and only if

$$K=K_1+\cdots+K_m,\qquad L=L_1+\cdots+L_m,$$

for compact convex sets $K_1,\ldots,K_m,L_1,\ldots,L_m$ of dimension at least one, with $\sum_{i=1}^m\dim K_i=n$, and $K_i$ and $L_i$ homothetic for $i=1,\ldots,m$.