10 problems

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 $K\subset\mathbb{R}^n$ be a convex body with $n\geq 3$, and let $p\in\mathbb{R}^n$. Assume that every $(n-1)$-dimensional section of $K$ through $p$ is centrally symmetric, and…

Arnold's conjecture. Algebraically integrable domains with $C^{\infty}$ boundaries do not exist when $n$ is even, and when $n$ is odd all such domains are ellipsoids.

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 $K_1\subsetneq K_2\subsetneq K_3\subset\mathbb R^n$, $n\geq 2$, be pairwise different convex bodies. A body of flotation for a convex body is the corresponding notion used in t…

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…

A3's conjecture. In any dimension, $K$ is an ellipsoid if and only if there exists some $m\in\mathbb N$ such that $A_K^m(\xi,t)$ is a polynomial in $t$.

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…

Let $K\subset\mathbb R^n$ be a convex body, with $n\geq 3$, and let $p\in\mathbb R^n$. Suppose that all $2$-dimensional sections of $K$ through $p$ have a centre of symmetry, and t…