12 problems

Let $B_n$ be the polytope of $n\times n$ doubly stochastic matrices, and let $e(B_n,t)$ denote the number of lattice points in $tB_n$. The dimension of $B_n$ is $(n-1)^2$. For a la…

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 $S$ be a closed piecewise-linear surface, with facets indexed by $i$, unit facet normals $n_i$, facet volumes $V_i$, and dihedral angles $\theta_{FG}:=\angle(n_F,n_G)$. Conside…

McMullen–Schneider–Shephard conjecture. Every polytope with the generating property is strongly monotypic in every dimension.

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 $P$ be a simplicial polytope, let $P^*$ be its dual polytope, and let the adjoint of $P$ be the polynomial appearing as the numerator of the normalized generating function for…

Let $\mathcal{A}$ be a set of vertices, meaning points that are extrema of their convex hull, and suppose that a new vertex is added while all previous points remain vertices. Pyra…

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…

Amoeba conjecture. $U_n$ is the $2^n$-th part of the amoeba of a certain Laurent polynomial, and the Newton polytope of this polynomial is the union, with the central point…

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…

Let $d$ be a positive integer and let $P$ be a convex polytope in $\mathbb R^d$. An orthoscheme is a simplex whose successive edges are pairwise orthogonal in the standard ordering…