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.

Problem 4.1.* Let $\Omega \subset \mathbb{R}$ be a finite union of intervals and $\nu$ a weak tiling measure for $\Omega$. Must $\mathrm{supp}(\nu)$ have bounded density?

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 $\gamma_n(C) \geq 0.01$?

Problem 4.2.* Let $\Omega \subset \mathbb{R}$ be a finite union of three or more intervals. If $\Omega$ weakly tiles its complement, must it also tile its complement properly?

Problem 4.3.* Let $\Omega \subset \mathbb{R}$ be a finite union of intervals and $\nu$ a weak tiling measure for $\Omega$. Must $\nu$ be expressible as a convex combination of proper tiling measures?

Fuglede's conjecture* in one dimension: A bounded subset of ℝ with positive Lebesgue measure is spectral iff it tiles ℝ by translation.

Fuglede's conjecture* in two dimensions: A bounded subset of ℝ^2 with positive Lebesgue measure is spectral iff it tiles ℝ^2 by translation.