6 problems

Let $T(d,k)$ be a constant such that, for every set of $n$ points with $k$ independent and $d-k$ dependent degrees of freedom, there is a $k$-flat $F$ and a partition of the points…

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 $n$ points be given in $\mathbb{R}^d$, and measure the quality of a regression hyperplane by its regression depth. Rousseeuw and Hubert's conjecture asserts that a regression h…

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 $\operatorname{RBP}$ denote the regression breakdown point, and let (B) be the result in the paper giving an upper bound for it. Sharp lower-bound conjecture. We conjecture tha…