28 problems

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, and let $K,L$ be origin-symmetric convex bodies in $\mathbb{R}^n$. For every $\lambda\in[0,1]$, Gardner–Zvavitch…

Let $K,L\subset\mathbb R^n$ be symmetric convex sets, let $\lambda\in[0,1]$, and let $p\in[0,1]$. For $p>0$, define the $L_p$-sum by … with the limiting case $p=0$ given by the zer…

Nonabelian Brunn–Minkowski conjecture. For every pair of compact sets $A,B\subseteq G$,

Let $cmu$ be an even log-concave measure on $cmathbb{R}^d$, let $clambdacin(0,1)$, and let $K,Lcsubsetcmathbb{R}^d$ be convex and centrally symmetric. Gardner–Zvavitch's dimensiona…

Böröczky–Lutwak–Yang–Zhang's Log-Brunn–Minkowski conjecture. The inequality

Let $K\subset\mathbb{R}^n$ be a convex body, let $-K=\{-x:x\in K\}$ be its reflection through the origin, let $\operatorname{vol}_n$ denote $n$-dimensional volume, and let $A+B$ de…

A measure $\mu$ on $\mathbb{R}^n$ is log-concave if, for every convex compact sets $K,L\subset\mathbb{R}^n$ and every $t\in[0,1]$, it satisfies … The measure is even when…

$L_p$-Brunn–Minkowski conjecture. Does it necessarily follow that

Let $d,k4in\mathbb{N}$ and let $\mathcal{L}_1,\dots,\mathcal{L}_k\in\operatorname{Mat}_d(\mathbb{Z})$ be irreducible and coprime. Let $A\subseteq\mathbb{Z}^d$ be finite and non-emp…

Let $A\subset\mathbb{R}^n$ be measurable and let $\delta<\Delta$. Locality conjecture. There is an absolute constant $\Delta>0$ such that if … then there exists a convex set…

Let $n\in\mathbb{N}$ and $t\in(0,1/2]$, and let $A,B\subset\mathbb{R}^n$ be measurable sets of equal volume satisfying … where $\delta\in[0,d_{n,t}]$. Optimal parameter-dependence…

Let $n\in\mathbb{N}$, let $t\in(0,1/2)$, and let $A,B\subset\mathbb{R}^n$ be measurable sets of equal measure satisfying … where $\delta<d_{n,t}$. Linear convex-hull stability conj…

Let $n\in\mathbb{N}$ with $n\geq 2$, let $t\in(0,1/2)$, and let $A,B\subset\mathbb{R}^n$ be measurable sets of equal measure satisfying … where $\delta<d_{n,t}$. Square-root stabil…

Let $\Delta>0$ be an absolute constant. Convex approximation conjecture. If $\delta<\Delta$ and $A\subset\mathbb{R}^k$ satisfies … then there is a convex set $K\subset\mathbb{R}^k$…

Let $\mu$ be an even log-concave measure on $\mathbb{R}^n$, let $\lambda\in(0,1)$, and let $K$ and $C$ be origin-symmetric convex bodies. Colesanti–Livshyts–Marsiglietti conjecture…

Let $\lambda\in(0,1)$, let $K$ and $C$ be origin-symmetric convex bodies in $\mathbb{R}^n$, and let $\gamma_n$ be the canonical Gaussian probability measure. Gardner–Zvavitch conje…

Dar's conjecture. Dar conjectured that

Let $K$ and $C$ be convex bodies in $\mathbb{R}^n$, and let $\gamma^*(n)$ denote the dimensional stability constant in the Figalli–Maggi–Pratelli stability inequality for the Brunn…

A Borel measure $\mu$ on $\mathbb{R}^{n}$ satisfies the (B) property if, for every symmetric convex set $K\subset\mathbb{R}^{n}$, the function … is log-concave on $\mathbb{R}$. Her…

Let $p\in[0,1]$, let $\mu$ be an even log-concave measure on $\mathbb R^n$, let $K,L$ be nonempty symmetric convex sets, and let $\lambda\in[0,1]$. For $p>0$, define … with $p=0$ i…

For finite two-dimensional sets $A,B one$ in the plane that are not collinear, let $[A]$ and $[B]$ denote their convex hulls, and let ${rm tr}(A)$ and ${rm tr}(B)$ be the numbers o…

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, defined by … For $\lambda\in(0,1)$ and sets $A,B\subset\mathbb{R}^n$, write…

Dar's conjecture. The Brunn–Minkowski inequality should satisfy

Let $n\geq 2$, and let $C(K,n)$ and $C(n)$ denote the optimal constants in the quantitative anisotropic isoperimetric inequality and quantitative Brunn–Minkowski inequality, respec…

Sharp quantitative Brunn–Minkowski stability conjecture. For arbitrary sets $E$ and $F$, the optimal exponent of $\alpha(E,F)$ in a quantitative stability inequality for the Brunn–…