5 problems

Fradelizi--Meyer's conjecture. For every convex body $K\subset\mathbb{R}^n$,

Let $K$ be a convex body in $\mathbb{R}^{n}$, and let $(K-K)^{\circ}$ be the polar body of its difference body. Makai Jr.'s conjecture. … Equality should hold if and only if $K$ is…

Let $K$ be a centrally symmetric convex body in $cmathbb{R}^{n}$, and let $\mathcal{P}(K)$ denote its volume product. Let $B_{\infty}^{n}$ be the unit ball of the $\ell_{\infty}$ n…

Volume-product conjecture for finite metric spaces. One has

Let $(X,\|\cdot\|)$ be an $n$-dimensional normed space, let $(X^*,\|\cdot\|^*)$ be its dual, and let $B$ and $B^{\circ}$ be the unit balls of $X$ and $X^*$, respectively. Write…