7 problems

Let $k\in\mathbb{N}$ and let $\epsilon$ be positive. For finite sets $A,B\subset\mathbb{Z}^k$, say that $B$ is not covered by $n$ parallel hyperplanes when no collection of $n$ par…

Functional $L_p$-Gardner–Zvavitch conjecture. There exists an absolute constant $C\geq1$ such that

Let $N\geq 2$ and let $\alpha\in(0,N)$. For a convex body $K\subset\mathbb{R}^N$, write ${\rm Cap}_{\alpha}(K)$ for its $\alpha$-Riesz capacity. For convex bodies $K_0,K_1$ and…

Let $K$ and $L$ be symmetric convex bodies in $\mathbb{R}^n$, let $V$ denote volume, and let $\lambda\cdot K+_p(1-\lambda)\cdot L$ denote their $L^p$-Minkowski combination. The sym…

The unconditional log-Brunn–Minkowski conjecture. Inequality $V(\lambda\cdot K+_o(1-\lambda)\cdot L)\geq V(K)^\lambda V(L)^{1-\lambda}$ is true for all $\lambda\in[0,1]$ and all sy…

Let $A$ and $B$ be finite non-collinear point sets in $\mathbb{R}^2$. For a finite non-collinear set $S$, let $b_S$ and $i_S$ denote the numbers of points of $S$ on the boundary an…

Brunn–Minkowski equality conjecture. Equality holds if and only if $\Omega_0$ and $\Omega_1$ are homothetic. This is the equality case left open for the Brunn–Minkowski inequality…