Convex bodies off center (Q2650281)

Let \(0 < t < 1\), \(n\ge 2\), and let \(F(\mathfrak x)\) be the distance function of a symmetric convex body \(K\) in \(R_n\); suppose that \(\mathfrak x=\mathfrak o\) is the only vector with integral coordinates satisfying \(F(\mathfrak x) \le 2t J^{-1/n}\) where \(J\) is the volume of \(K\). Then to every vector \(\mathfrak a\) there is an integral vector \(\mathfrak g\) satisfying \[ F(\mathfrak a - \mathfrak g) < [1 + (n-1) t^n] t^{1-n} J^{-1/n}.\] This further improves results by the reviewer [Proc. Akad. Wet. Amsterdam 41, 634--637 (1938; Zbl 0019.05105)] and by \textit{H. Störmer} and \textit{G. Walter} [Arch. Math. 2, 346--348 (1950; Zbl 0040.16402)].