Stretching convex domains to capture many lattice points

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.