5 problems

Let $n\ge 2$, and let $F_n$ be the $n\times n$ matrix whose $(i,j)$ entry is $1$ when $j\le i+1$ and $0$ otherwise. The matrix $F_n$ is a union of permutation matrices correspondin…

Let $S(t)$ be a differentiable one-parameter family of oriented piecewise-linear surfaces. For each facet $F$, let $n_F$ be its unit normal and $V_F$ its volume, and define the dih…

Let $\Omega_1(d,n)$ be the polytope of $1$-polystochastic $d$-dimensional matrices of order $n$, and let a local extremum mean a local maximum or local minimum of the permanent on…

Let a center assign a point to every simplicial polytope in ${\mathbb R}^n$. Assume that the center depends analytically on the polytope, commutes with dilations, and satisfies the…

Let $n$ be a positive integer, let $0<q<1$ and $t>0$, and let ${{\text{\hskip.03cm\bf{T}}}}_n(q,t)$ denote the Tutte polytope defined in the source. Conjectured face numbers of the…