2 problems

Let $S\subset \mathbb{R}^{n}$ be an embedding of $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$ that is $O$-star, and let $\phi:S\to S$ map each point to the unique point on the ray from…

Let $K,L\subset\mathbb R^n$ be convex bodies with $L\subset\operatorname{int}K$. For each $x\in\partial K$, let $\Sigma(L,x)$ denote the graze of $L$ from $x$, namely the intersect…