In a classic paper, 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 $$\left|x_1\right|\left({\left|x_1\right|}^3+{\left|x_2\right|}^3+{\left|x_3\right|}^3\right)\le 1.$$ 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...