5 problems

Let $K$ and $L$ be convex bodies in $\mathbb{R}^n$. For $\lambda\in(0,1)$, their geometric mean is defined, for origin-symmetric bodies, by … Here $V$ denotes volume,…

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 $\nu$ 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 $p \in [0,1]$. 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…

Let $K,L$ be symmetric convex bodies in $\mathbb R^n$. For $\lambda\theta \in [0,1]$, define their logarithmic Minkowski combination by … where $h_K$ and $h_L$ are support function…