  Suppose we count the positive integer lattice points beneath a convex
decreasing curve in the first quadrant having equal intercepts. Then stretch in
the coordinate directions so as to preserve the area under the curve, and again
count lattice points. Which choice of stretch factor will maximize the lattice
point count? We show the optimal stretch factor approaches $1$ as the area
approaches infinity. In particular, when $0<p<1$, among $p$-ellipses
$|sx|^p+|s^{-1}y|^p=r^p$ with $s>0$, the one enclosing the most first-quadrant
lattice points approaches a $p$-circle ($s=1$) as $r \to \infty$.
  The case $p=2$ was established by Antunes and Freitas, with generalization to
$1<p<\infty$ by Laugesen and Liu. The case $p=1$ remains open, where the
question is: which right triangles in the first quadrant with two sides along
the axes will enclose the most lattice points, as the area tends to infinity?
  Our results for $p<1$ lend support to the conjecture that in all dimensions,
the rectangular box of given volume that minimizes the $n$-th eigenvalue of the
Dirichlet Laplacian will approach a cube as $n \to \infty$. This conjecture
remains open in dimensions four and higher.
