7 problems

Let $n\geq 2$ be an integer. Let $K$ and $L$ be symmetric convex bodies in $\mathbb R^n$ containing the origin in their interiors. For $\lambda\in[0,1]$, define their geometric ave…

Lutwak's conjecture. The inequality

Let $\Omega_0$ and $\Omega_1$ be open, bounded, centrally symmetric subsets of $\mathbb{R}^n$, and let $t\in[0,1]$. Write … The Gaussian torsional rigidity of a set $\Omega$ is den…

Let $\Omega_0$ and $\Omega_1$ be convex sets containing the origin, and let $\Omega_t=(1-t)\Omega_0+t\Omega_1$ for $t\in[0,1]$. The Gaussian measure is denoted by $\gamma$. Garcia-…

Let $\nu$ be an even log-concave measure on $\mathbb{R}^{n}$. Let $K_{0},K_{1}\subseteq\mathbb{R}^{n}$ be convex and origin-symmetric, meaning $K_{0}=-K_{0}$ and $K_{1}=-K_{1}$. Ev…

Let $n\geq 2$. Let $\gamma$ be a log-concave symmetric measure on $\mathbb R^n$, meaning that $\gamma(A)=\gamma(-A)$ for every measurable set $A$. Let $K$ and $L$ be symmetric conv…

Let $p \in [0,1]$. Let $\mu$ be a symmetric measure in $\mathbb R^n$ with an $\alpha$-concave density function, where $\alpha \geq -\frac{p}{n}$. Let $\oplus_p$ denote the $p$-Fire…