12 problems

Let $A\subseteq\mathbb{R}^n$ be a general measurable set, let $B_2^n$ denote the Euclidean unit ball, and let $t\geq 0$ be a dilation parameter. Costa–Cover conjecture. The volume…

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…

Let $A$ and $B$ be homothetic regions, and let $\tau$ and $\delta$ denote the parameters measuring the relative size and Brunn–Minkowski deficit introduced in the paper. Let…

Let $K$ and $C$ be convex bodies in $\mathbb{R}^n$, and let $\gamma^*(n)$ denote the largest constant for which the quantitative Brunn–Minkowski inequality … holds for all such $K$…

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

Let $K$ and $L$ be convex bodies, and let $z_K$ and $z_L$ denote the translation vectors appearing in the log-Brunn-Minkowski inequality. The centroid of a convex body is its cente…

Let $E,K\subset\mathbb{R}^n$ be the sets and $c_n$ the constant appearing in the quantitative isoperimetric estimate … where $\alpha(E)$ denotes the asymmetry used in that estimate…

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…

Let $\Phi$ be a continuous translation invariant and $\mathrm{SO}(n)$-equivariant Minkowski valuation of degree $j$, where $j\in\{2,\ldots,n-1\}$. For $K,L\in\mathcal{K}^n$, let…

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…