9 problems

Let $K,L$ be convex bodies in $\mathbb{R}^n$. For $t\in[0,1]$, let $K^{1-t}L^t$ denote the logarithmic combination defined in the paper. A convex body is symmetric when it satisfie…

Bolker's conjecture. Every zonoid of dimension at least $3$ whose polar is also a zonoid must be an ellipsoid.

Improved concavity conjecture. The function $P^{1/m}(t)$ should be concave for $t\geq 0$.

Let $A$, $B_1$, and $B_2$ be zonoids in ${\mathbb R}^n$. Plünnecke–Ruzsa conjecture for zonoids. One has … This is a log-submodularity conjecture for volume under Minkowski additio…

Let $n\geq1$ and let $A,B,C$ be zonoids, hence convex bodies in $\mathbb{R}^n$. Zonoid supermodularity conjecture. … This is motivated by the proved case in which the first summand…

Let $K,L\in\mathcal{Z}(\mathbb{R}^m)$ be zonoids and let $C\in\mathcal{G}(m-2,\mathbb{R}^m)$ be a Grassmannian zonoid of degree $m-2$. Conjecture. For any…

Let $K$ be a zonoid in $\mathbb{R}^d$, let $V(K)$ be its volume, and let $\Pi^2K:=\Pi(\Pi K)$ be its second projection body. Saroglou's upper-bound conjecture. … with equality if a…

Let $P(K):=V(\Pi K)/V(K)^{d-1}$, and restrict $K$ to zonoids. A Weil body is one of the bodies described in the source as a Cartesian product of symmetric convex bodies of dimensio…

Let $K$ be a convex body in ${\mathbb{R}}^n$. Its intersection body $I K$ is the body whose radial function satisfies … A convex body is a polar zonoid if its polar body is a zonoi…