5 problems

Let $\mathcal{C}(K)=\lvert K\rvert^2/\det\operatorname{Cov}(K)=1/L_K^{2n}$ for a convex body $K\subseteq\mathbb{R}^n$, and let $\Delta_{n,0}$ be the standard simplex. The strong is…

Let $\mathcal{C}(K)=\lvert K\rvert^2/\det\operatorname{Cov}(K)=1/L_K^{2n}$ for a convex body $K\subseteq\mathbb{R}^n$. The strong isotropic constant conjecture for symmetric convex…

For a convex body $K\subseteq\mathbb{R}^n$, let $L_K$ denote its isotropic constant. The isotropic constant conjecture. There is a universal constant $c>0$ such that, for every…

For $u\in\partial B_2^n$, define $u^\perp=\{x\in\mathbb{R}^n:\langle x,u\rangle=0\}$. For a convex body $K\subseteq\mathbb{R}^n$, let $b(K)=\int_K x\,dx/\lvert K\rvert$ be its bary…

Let $G\subset\mathbb{R}^d$ be a convex symmetric body in isotropic position, and let $L(G)$ denote its isotropic constant. Slicing conjecture. There is a constant $C>0$ independent…