7 problems

Let $n>k\geq 1$, let $\Box^n=[-1,1]^n$, and let $H$ be a $k$-dimensional linear subspace. Write $C_{\Box}(n,k)2^k$ for the maximum volume among sections $\Box^n\cap H$ that are aff…

Two-block critical-direction conjecture. Based on numerical evidence, there are no non-diagonal critical directions of the form $\mathbf{v}_{n,k}(a)$ with $3\leq k\leq n-2$.

Diagonal extremality conjecture. All locally extremal central sections of $Q_n$ are diagonal for each $n\geq 2$.

Let $C^n=[-\frac 12,\frac 12]^n$ be the unit cube in $\mathbb{R}^n$, and for $u\in\mathbb{R}^n$ let $H(u)=u^\perp$ be the hyperplane through the origin orthogonal to $u$. Write…

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…

Bárány–Frankl conjecture. The maximum in $V_d$ is attained by ${\boldsymbol v}={\boldsymbol 1}_d=(1,\ldots,1)$.

Let $\Box^n=[-1,1]^n$ be the standard cube, and let $H_k$ be a $k$-dimensional subspace of $\mathbb{R}^n$. Define $C_{\Box}(n,k)2^k$ to be the maximum volume among sections…