5 problems

Let $K\subset \mathbb{R}^{n}$ be a convex body with $n\geq 3$, and let $p\in\operatorname*{\int}K$. For every hyperplane $\Pi$ passing through $p$, suppose the section $\Pi\cap K$ h…

Let $K\subset \mathbb{R}^3$ be a convex body. A plane section means the intersection of $K$ with an affine plane, and an axis of symmetry is a line in that plane whose reflection p…

Let $K_1,K_2\subset\mathbb R^n$, $n\geq 2$, be different convex bodies. For every affine hyperplane $H\in\operatorname{Graff}_{n-1}(\mathbb R^n)$ that intersects both bodies, consi…

Let $K\subset \mathbb{R}^{n}$, $n\geq 3$, be a convex body. A point $p\in\mathbb{R}^{n}$ is a Larman point of $K$ if, for every hyperplane $\Pi$ passing through $p$, the section…

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