7 problems

Let $K\subset\mathbb R^n$ be a convex body, and for $m\geq 1$ define … Set … In the range $n\geq3$ and $m\geq2$, Schneider's conjecture. … with equality if and only if $K$ is an el…

Let $K\subset\mathbb R^n$ be a convex body, and let $D^m(K)\subset\mathbb R^{nm}$ denote its $m$th-order difference body. For fixed volume $|K|_n$, Schneider conjectured that the v…

Let $K\subset\mathbb{R}^n$ be a convex body, let $\lambda\in(0,1)$, and define the unbalanced difference body … Let $\Delta$ be an $n$-dimensional simplex. Unbalanced difference-bo…

The simplex extremal conjecture. If $\Delta$ is an $n$-dimensional simplex, then

Ellipsoid containment conjecture. There exists an ellipsoid $E\subset\mathbb{R}^n$ such that

Difference-body volume-product conjecture. The sharp lower bound is

Let $K$ be an $n$-dimensional convex body and let $\mu,\nu$ be nonnegative real numbers. Write $\mu K-\nu K=\{\mu x-\nu y:x,y\in K\}$. Simplex extremal conjecture. For every $n$-di…