13 problems

Kolesnikov–Milman first even eigenvalue conjecture. The infimum of the first even eigenvalue is attained by the cube:

Kolesnikov–Milman local $L_p$-Minkowski conjecture. For every $z\in C^2_e(S^n)$ satisfying

Even $L_p$-Minkowski uniqueness conjecture. For $p\in[0,1)$, the Monge–Ampère equation

Böröczky–Lutwak–Yang–Zhang conjecture. The $L_p$ Brunn–Minkowski inequality holds:

Critical integrability conjecture. The surface area measure of every $C$-asymptotic set satisfies

Log-Minkowski conjecture.

Let $K,L\subset\mathbb H^{n+1}$ be smooth origin-symmetric, uniformly h-convex bounded domains, with $n\geq1$. Let $\varphi_K,\varphi_L$ be their horospherical support functions, l…

Let $n\geq1$, let $0\leq k\leq n$, and let $K,L\subset\mathbb{H}^{n+1}$ be smooth uniformly h-convex bounded domains. Write $\varphi_K,\varphi_L$ for their horospherical support fu…

Bianchi–Böröczky–Colesanti–Yang conjecture. There exists a convex body $K$ in $\mathbb{R}^n$ such that

Let $\mathcal{K}^{2,\alpha}_{+,e}$ be the origin-symmetric convex bodies with $C^{2,\alpha}$-smooth boundary and strictly positive curvature. For $K\in\mathcal{K}^{2,\alpha}_{+,e}$…

Strict even spectral-gap conjecture. For every $K\in\mathcal{K}^2_{+,e}$,

Characterization conjecture. The measure $\mu$ is the $L_p$-surface area measure of a convex body $K\in\mathcal K_0^n$ if and only if $\operatorname{supp}\,\mu$ is not a pair of an…

Lutwak–Yang–Zhang's conjecture. Under these assumptions, $\mathbb K$ is a Euclidean ball.