  The Heilbronn triangle problem asks for the placement of $n$ points in a unit
square that maximizes the smallest area of a triangle formed by any three of
those points. In $1972$, Schmidt considered a natural generalization of this
problem. He asked for the placement of $n$ points in a unit square that
maximizes the smallest area of the convex hull formed by any four of those
points. He showed a lower bound of $\Omega(n^{-3/2})$, which was improved to
$\Omega(n^{-3/2}\log{n})$ by Leffman.
  A trivial upper bound of $3/n$ could be obtained, and Schmidt asked if this
could be improved asymptotically. However, despite several efforts, no
asymptotic improvement over the trivial upper bound was known for the last $50$
years, and the problem started to get the tag of being notoriously hard.
Szemer{\'e}di posed the question of whether one can, at least, improve the
constant in this trivial upper bound. In this work, we answer this question by
proving an upper bound of $2/n+o(1/n)$. We also extend our results to any
convex hulls formed by $k\geq 4$ points.
