Compositions of sums of absolute powers (Q1900088)

The author introduces oddballs \(G\subset \mathbb{R}^n\) defined by \(G= \{x\in \mathbb{R}^n \mid f(x)\leq 1\}\), where \(f: \mathbb{R}^n\to \mathbb{R}\) is a suitable sum of powers of \(x_1, \dots, x_n\). For a wide class of oddballs the author gives explicit formulas for their volume and their lattice-point-number. The motivation is the rather general principle that \(O\)-symmetric convex bodies with unusually few lattice points have an unusually large lattice packing density, exceeding the Minkowski-Hlawka bound. So there is a relation to the well-known dense packings of superballs by the same author, although the class of superballs differs from that of oddballs.