Abstract: Given an equivalence relation on finite point configurations, a typical question in extremal discrete geometry asks the maximum size $f_d(n)$ of an equivalence class and the minimum number $g_d(n)$ of distinct equivalence classes determined by $n$ points in $d$-space. In this talk, the finite point configurations are simplices, and two simplices are considered to be equivalent if they have the same volume. I'll show how incidence bounds, allowable sequences, and the crossing lemma in topological graph theory lead to new upper and lower bounds on $f_d(n)$ and $g_d(n)$ for $d=2$ and 3.

In the plane, every $n$ points determine at most $O(n^{44/19}) = O(n^{2.3158})$ unit triangles. This improves earlier bounds of Pach and Sharir from 1992. No better lower bound is known than $\Omega(n2\log \log n)$, due to Erd\H{o}s and Purdy from 1971, which is attained on the $\sqrt{\log n}\times (n/\sqrt{n})$ integer lattice section. Burton, Purdy, and Straus conjectured that every $n$ noncollinear points in the plane determine at least $\lfloor (n-1)/2\rfloor$ distinct triangle areas, which is attained by $n$ points distributed evenly on two parallel lines. Using allowable sequence and linear optimization, one can prove a lower bound of $.4458n-O(1)$.

In three-space, every $n$ points determine at most $O(n^{7/2})$ tetrahedra of unit volume, and there are point sets that determine at least $\Omega(n3 \log \log{n})$. The tetrahedra spanned by $n$ points in 3-space, not all on a plane, have at least $\Omega(n)$ distinct volumes, which is the best possible bound. This answers an old question of Erd\H{o}s, Purdy, and Straus.

(Joint work with Adrian Dumitrescu.)