6 problems

Let $K$ be a convex body and $L$ a convex body defining the $L$-convex floating body and wet part. In the planar case with $L=RB^2$, suppose that the Bárány–Larman-type relation fo…

Let $K,L\subset\mathbb R^n$, with $n\geq 3$, be convex bodies such that $L\subset\operatorname{int}K$ and $L$ is strictly convex, meaning that its boundary contains no segments. Fo…

Spherical floating-body conjecture. If $K_\delta$ is a Euclidean ball, then $K$ is a Euclidean ball.

The Homothety Conjecture. The homothety $K_\delta=t(\delta)K$ occurs only when $K$ is an ellipsoid.

Let $K$ be a convex body, and let its floating body and polar be considered together with the illumination body of the polar. Suppose that the separation body is constructed using…

Let $S\subseteq\mathbb{R}^n$ be a centrally symmetric convex body with $C_+^2$-boundary. For sufficiently small $\delta>0$, let $S_\delta$ denote its floating body, and for…