Open-problem statements, with their sources attached.

The Banach--Mazur rotation problem asks whether every separable Banach space whose group of linear isometric equivalences acts transitively on the unit sphere is linearly isometric to a Hilbert space.

Let $K \subset \mathbb{R}^n$ be a balanced compact set (that is, $\lambda K \subseteq K$ whenever $|\lambda| \leq 1$) and suppose that the normalised Gaussian measure $\gamma_n(K) \geq 0.99$. Does $10K$ contain a compact convex set $C$ with $\gamma_n(C) \geq 0.01$?