3 problems

Let $K,L\subset\mathbb R^n$, $n\geq 3$, be convex bodies with $L\subseteq K$. Two-body flat-shadow conjecture. If for every $p\in\partial K$ there are at least $L(n)$ point light s…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body with boundary of class $C^3$. Local light-source conjecture. If for every $p\in\partial K$ there are at least $n+1$ point lig…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body and let $S$ be a hypersurface that is the image of an embedding of the sphere $\mathbb S^{n-1}$, with $K$ contained in the in…