## Abstract

In a classic paper [14], W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt’s Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body

|x_{1}|(|x_{1}|^{3} + |x_{2}|^{3} + |x_{3}|^{3} ≤ 1.

In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result is improved and extended to the star body

|x_{1}|(|x_{1}|^{3} + |x_{2}^{2} + x_{3}^{2})^{3/2}≤ 1.

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