4 problems

$k$-Roudneff's conjecture. For any rank $r\geq 2k+1\geq 3$ oriented matroid $\mathcal{M}$ on $n\geq r+1$ elements,

Let $s(d,n)$ denote the number of simplicial $(d-1)$-spheres with $n$ labeled vertices, and let $\operatorname{sn}(d,n)$ denote the number of $\lfloor d/2\rfloor$-neighborly simpli…

Let $P$ be a $d$-dimensional convex polytope, and let $f_i(P)$ denote the number of its $i$-dimensional faces. A polytope is $k$-neighborly if every subset of $k$ vertices is the v…

A polytope is neighborly if every set of at most half the dimension's vertices forms a face, and a polytope is circumscribable if it has a realization whose facets are tangent to a…