11 problems

Let $K,L\in\mathcal{K}_e(\mathbb{R}^n)$ be origin-symmetric convex bodies containing the origin in their interiors. Let $V$ denote the mixed volume functional, let $S_K$ be the sur…

Let $K,L\in\mathcal{K}_e(\mathbb{R}^n)$ be origin-symmetric convex bodies containing the origin in their interiors, and let $K^tL^{1-t}$ denote their logarithmic combination, defin…

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

Let $n\geq1$ and let $p\in(n-2,n)$. For compact sets $A,B\subset\mathbb R^n$ and $\lambda\in[0,1]$, write … Let $\operatorname{Cap}_p$ denote the Riesz $p$-capacity. Novaga–Ruffini…

Let $C,D\in\mathrm{Conv}_s(\mathbb{R}^{n+1})$ be centrally symmetric convex bodies of unit volume, and let $h_C,h_D$ denote their support functions. Let $\nu_C$ be the cone measure…

Let $n\geq 1$ and $0\leq k\leq n$ be integers, let $\frac12\leq p\leq 2$, and let $a,b\geq 0$ satisfy $a+b\geq 1$. For smooth uniformly h-convex bounded domains…

Let $k\geq 2$, and let $A_1,A_2,\ldots,A_{k+1}$ be compact sets in $\mathbb{R}^d$. For a finite family of sets, $\sum_i A_i$ denotes their Minkowski sum, and $\operatorname{vol}$ d…

The local $L^p$-Brunn-Minkowski inequality. One has

The log-Brunn-Minkowski conjecture. One has

The $L^p$-Brunn-Minkowski conjecture. One has

Eventual concavity conjecture. There exists $t_0$ such that $V_{A,B}(t)$ is $\frac{1}{n}$-concave on $[t_0,+\infty)$. This is proposed as a weaker form of the Costa–Cover conjectur…