15 problems

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…

Let $\nu$ be an even log-concave measure on $\mathbb{R}^{n}$. For origin-symmetric convex bodies $K_{0},K_{1}\subseteq\mathbb{R}^{n}$ and $t\in[0,1]$, define … The measure $\nu$ is…

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 $n\in\mathbb{N}$ 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…

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

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 $p\in[0,1]$, 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…

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…

Let $S(n)$ be the symmetric group equipped with the metric $\mathrm{d}$, and let $A,B\subseteq S(n)$ be nonempty. Write $M$ for their midpoint set, and let $\mathrm{d}(A,B)$ denote…

Let $A$ be a bounded measurable set in $cmathbb{R}^n$, let $B_2^n$ denote the Euclidean closed unit ball, and let $|\cdot|$ denote Lebesgue measure. Costa–Cover conjecture. The fun…

The fractional Brunn–Minkowski conjecture. For every such fractional partition,