5 problems

Let $S$ be a set of $n$ points in $d$-dimensional space. The regression depth of a hyperplane is the minimum number of points intersected by the hyperplane as it undergoes a contin…

Let $d$ and $0\leq k<d$ be constants. For a set of points with $k$ independent and $d-k$ dependent degrees of freedom, let $R(d,k)$ denote a constant such that some $k$-flat has re…

Let $P$ be a set of $n$ points, and let regression depth be the minimum number of points intersected by a hyperplane during any continuous motion taking it to a vertical hyperplane…

Let $1\leq k\leq d$ be integers, and let $A_1,\dots,A_{d-k+1}$ be finite arrangements of hyperplanes in $\mathbb{R}^d$. Assume that … for positive integers $r_i$, for each…

Let $r,d$ be positive integers, and let $A_1,\dots,A_{d+1}$ be sets of $r$ hyperplanes each in $\mathbb{R}^d$. A partition of their union into $r$ sets $B_1,\dots,B_r$ is colorful…