  We consider an optimal stretching problem for strictly convex domains in
$\mathbb{R}^d$ that are symmetric with respect to each coordinate hyperplane,
where stretching refers to transformation by a diagonal matrix of determinant
$1$. Specifically, we prove that the stretched convex domain which captures the
most positive lattice points in the large volume limit is balanced: the
$(d-1)$-dimensional measures of the intersections of the domain with each
coordinate hyperplane are equal. Our results extend those of Antunes & Freitas,
van den Berg, Bucur & Gittins, Ariturk & Laugesen, van den Berg & Gittins, and
Gittins & Larson. The approach is motivated by the Fourier analysis techniques
used to prove the classical $\#\{(i,j) \in \mathbb{Z}^2 : i^2 +j^2 \le r^2 \}
=\pi r^2 + \mathcal{O}(r^{2/3})$ result for the Gauss circle problem.
