10 problems

Mahler's conjecture. For every convex body $\Omega\subset\mathbb{R}^N$,

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 convex body in $\mathbb{R}^{n}$, and let $\mathcal{P}(K)$ denote its volume product. Let $\Delta_n$ be a regular simplex in $\mathbb{R}^{n}$. Non-symmetric Mahler's co…

Volume-product conjecture for finite metric spaces. One has

For a graph $G$, let $\overline{G}$ be its complement, let $P_G$ be the associated graph polytope, and let $\mathsf{U}P_G$ denote the centrally symmetric polytope used in the paper…

Guggenheimer's conjecture. The minimal volume product $V(C)V(C^*)$ is attained when $C$ consists of the longest diagonals of a regular $(2k+1)$-gon with centre $0$, taken always in…

Orthogonal simplices conjecture. Among polytopes with at most $n+k$ vertices, the convex hull of $k$ simplices living in orthogonal subspaces of dimensions…

Fradelizi–Meyer's minimizer conjecture. The minimizing functions are conjectured to be, in a suitable system of coordinates with origin at $o$, the functions specified in the cited…

Difference-body volume-product conjecture. The sharp lower bound is

Saint Raymond's conjecture. The volume product $|K|\cdot|K^*|$ attains its minimum exactly for the Hanner-Hansen-Lima bodies.