A note on distinct volume subsets problem
For $2 \leq a \leq d+1$, what is the largest integer $H_{a,d} (n)$ such that every set of $n$ points in $\mathbb{R}^d$ with no $a$ points on a common $(a-2)$-flat contains a subset of $H_{a,d} (n)$ points whose determined $(a-1)$-dimensional simplices have pairwise distinct $(a-1)$-dimensional volumes? We construct $n$-point sets that improve the best known upper bounds for $H_{a,d}(n)$ in several cases of $a$ and $d$. We also study a dual version of the problem. Let $D_d(n)$ the maximum number such that for any arrangement of $n$ hyperplanes in general position in $\mathbb{R}^d$, we can always find a subset of $D_d(n)$ hyperplanes for which all the $d$-dimensional simplices that they define have distinct $d$-dimensional volumes. We improve the current known upper bound for $D_d(n)$ and give the first nontrivial lower bound for $D_2(n)$ and $D_3 (n)$.