  We aim to maximize the number of first-quadrant lattice points in a convex
domain with respect to reciprocal stretching in the coordinate directions. The
optimal domain is shown to be asymptotically balanced, meaning that the stretch
factor approaches 1 as the "radius" approaches infinity. In particular, the
result implies that among all p-ellipses (or Lam\'e curves), the p-circle
encloses the most first-quadrant lattice points as the radius approaches
infinity.
  The case p=2 corresponds to minimization of high eigenvalues of the Dirichlet
Laplacian on rectangles, and so our work generalizes a result of Antunes and
Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of
van den Berg, Bucur and Gittins.
  The case p=1 remains open: 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?
