8 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 $B_p^n$ denote the unit ball of $ell_p^n$, and let a hyperplane section be $B_p^n\cap H$ for a hyperplane $H$ through the origin. The diagonal directions are the hyperplanes wh…

Two-candidate conjecture. The maximal hyperplane section of $B_p^n$ is either given by $(a^{(2)})^\perp$ or by $(a^{(n)})^\perp$:

Let $Q_n=[-\frac12,\frac12]^n$ be the unit cube, and define … Here $|a|$ is the Euclidean norm and $\|a\|_1$ is the $\ell_1$ norm. Bárány–Frankl's conjecture. The maximum of…

Hyperplane-section conjecture. Under these conditions,

Let $Q^d=[0,1]^d$, let $e=e_1+dots+e_d$, and for a nonzero vector $v\in\mathbb{R}^d$ define … where $A(v,t)=\{x\in\mathbb{R}^d: v\cdot x=t\}$, and set $V_d=\max_{v\ne0}V_d(v)$. The…

Let $P=\mathbb{P}^N$, and let $X\subset P$ be a general complete intersection of multi-degree $(d_1,d_2,\ldots,d_k)$ with $d_1\leq\cdots\leq d_k$. Assume that $d_1\gg0$. For a hype…

Small-radius maximal-section conjecture. If $r$ is sufficiently small, then the section orthogonal to $a$ is maximal.