4 problems

Let $C \subseteq \mathbb R^n \times \mathbb R^d$ be a compact, convex set and let $S=C\cap(\mathbb Z^n\times\mathbb R^d)$. For $S$, define its mixed-integer volume by … with counti…

Let $P$ be a set of $n$ points in $\mathbb{R}^d$, and let $0\leq k\leq d$. For a point $q$, let $F_q$ be a $(d-k-1)$-flat through $q$, and let $F_o$ be such a flat disjoint from…

Let $P$ be a set of $n$ points in $\mathbb{R}^d$, and let $0\leq k\leq d$. A $k$-simplex is the convex hull of $k+1$ points of $P$, and a $(d-k)$-half flat is a half-flat of dimens…

Let $G_{\mathrm s}$ be the $n$-point stretched grid in $\mathbb{R}^d$, and let the centerflat depth conjecture assert a leading constant $\frac{k+1}{k+d+1}$ for the depth of a $k$-…