10 problems

Let $B_1^n=\operatorname{conv}\{\pm e_1,\ldots,\pm e_n\}$ be the cross-polytope, and let … For $a\in\mathcal{M}$, consider the central hyperplane section $B_1^n\cap a^\perp$. Maxim…

Let $n\ge 4$, and consider exotic flexible cross-polytopes in the sphere $\mathbb{S}^n$. These are flexible cross-polytopes whose dihedral-angle tangents have the exotic parametriz…

Let $\sigma=1$ denote the cross-polytope case among packings of superballs, and consider the Minkowski--Hlawka lower bound on lattice packing density. Rush's conjecture. No exponen…

Strong B-inequality conjecture for $B_p^n$ balls. This function is log-concave on $^n$ for every subspace $H$ of $^n$ and every $p>1$. The theorem stated in the paper establishes t…

Let $\Diamond^n$ be the standard cross-polytope in $\mathbb{R}^n$, let $H_k$ be a $k$-dimensional subspace, and let $\Diamond^n|H_k$ denote its projection onto $H_k$. Define…

Cross-polytope dual-pair conjecture. The dual pair $\mathcal U$ is isomorphic to a dual pair in the boundary complex of the $(d+1)$-dimensional cross-polytope.

Let $H_k$ be a $k$-dimensional subspace and let $\Diamond^n\mid H_k$ denote the corresponding affine section of the $n$-dimensional cross-polytope, while $\Diamond^k$ is the $k$-di…

Cross-polytope conjecture. Then $A_1,\ldots,A_{2n}$ are the vertices of an $n$-dimensional cross-polytope.

Let $K\subset\mathbb{R}^n$ be a convex body, let $V_i(K)$ denote its $i$th intrinsic volume, and let $r_1(K),\dots,r_n(K)$ be its successive inner radii. For…

Let $M^6_{16}$ be the constructed $16$-vertex combinatorial triangulation of $S^4\times S^2$, viewed as a $2$-Hamiltonian subcomplex of the $8$-dimensional cross polytope $\beta^8$…