3 problems

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 $P=P_0\cup\cdots\cup P_d\subset\mathbb{R}^d$ be partitioned into $d+1$ color classes, with the origin in the relative interior of every $\operatorname{conv}(P_i)$, and let colo…

For a finite set $P$ of $n$ points in $\mathbb{R}^d$, define its simplicial depth as the maximum number of $d$-simplices spanned by $P$ that contain a common point. Bukh–Matoušek–N…