24 problems

Let $K,L\subseteq\mathbb{R}^n$ be symmetric convex sets, and let their geometric mean be the set obtained by combining their support functions geometrically. Denote $n$-dimensional…

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…

Let $n\geq 2$, let $p\in[0,1)$, and let $\Omega_1,\Omega_2$ be origin-symmetric convex bodies in $\mathbb R^n$. For $t\in[0,1]$, write $(1-t)\cdot\Omega_1+_p t\cdot\Omega_2$ for th…

Let $K$ and $L$ be origin-symmetric convex bodies in $\mathbb{R}^n$, let $\lambda\in(0,1)$, and let $V$ denote volume. The logarithmic combination $(1-\lambda)K+_0\lambda L$ is the…

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 $K,L$ be convex bodies in $\mathbb{R}^n$. For $t\in[0,1]$, let $K^{1-t}L^t$ denote the logarithmic combination defined in the paper. A convex body is symmetric when it satisfie…

Let $K\in\mathcal{K}_{o}^n$ be a convex body and let $p>1$. The $L^p$ projection body ${\Pi}_pK$ and the $L^p$ centroid body ${\Gamma}_pK$ are defined by their support functions, a…

Let $\mu$ be an even log-concave probability measure on $\mathbb{R}^n$, and let $K,L$ be origin-symmetric convex bodies. The concavity exponent $p(\mu)$ is defined as the largest v…

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…

A symmetric set $K$ satisfies $K=-K$. Let $K$ and $L$ be symmetric convex bodies in ${\mathbb R}^n$, and let $t\in[0,1]$. Gardner–Zhang's conjecture. … This is a Brunn–Minkowski-ty…

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 $A$, $B_1$, and $B_2$ be zonoids in ${\mathbb R}^n$. Plünnecke–Ruzsa conjecture for zonoids. One has … This is a log-submodularity conjecture for volume under Minkowski additio…

Böröczky–Lutwak–Yang–Zhang conjecture. If $K$ and $L$ are origin-symmetric convex bodies in ${\mathbb R}^{n}$ and $0\leq\lambda\leq1$, then

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

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…

Kolesnikov–Milman's equivalence conjecture. For a given $p$, the local $L^p$-Brunn-Minkowski inequality is equivalent to the global $L^p$-Brunn-Minkowski inequality.

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

In phase space, consider coarse graining with convex polyhedra in an $L_p$ sense, and a Hamiltonian system with many degrees of freedom whose collective behavior is being described…

Cube minimizer conjecture. The cube minimizes this linearly invariant parameter:

Let $K\in\mathcal{K}^{2}_{+,e}$ be an origin-symmetric smooth strictly convex body, and let $L_K$ be its Hilbert–Brunn–Minkowski operator. Denote by $\lambda_{1,e}(-L_K)$ the first…

Let $n\geq 2$, $p\in[0,1)$, and let $K_0,K_1$ be origin-symmetric convex bodies in $\mathbb{R}^n$. For $\lambda\in[0,1]$, define the $L^p$-sum using the Aleksandrov body constructi…

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…