3 problems

Let $K\subset\mathbb{R}^2$ be a convex body with a center of symmetry and containing the origin, and let $p>1$. Planar symmetric-body equality conjecture. Equality in the relevant…

Let $K\subset\mathbb{R}^n$ be a convex body that contains the origin and has a center of symmetry, and let $p>1$. Let $q$ satisfy $1/p+1/q=1$. Symmetric-body Rogers--Shephard conje…

Let $B$ be an $n$-dimensional convex body, and let ${\mathcal I}(B)$ denote the smallest number of directions, or equivalently light sources, that illuminate $B$. A parallelotope h…