Geodesic
Estimating the critical determinants of a class of three-dimensional star bodies
Communications in Mathematics, Tome 25 (2017) no. 2, pp. 149-157.

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In the problem of (simultaneous) Diophantine approximation in~$\mathbb{R}^3$ (in the spirit of Hurwitz's theorem), lower bounds for the critical determinant of the special three-dimensional body $$ K_2:\quad (y^2+z^2)(x^2+y^2+z^2)\le 1 $$ play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant $\Delta (K_c)$ of more general star bodies $$ K_c:\quad (y^2+z^2)^{c/2}(x^2+y^2+z^2)\le 1, $$ where $c$ is any positive constant. These are obtained by inscribing into $K_c$ either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of $c$.
Classification : 11H16, 11J13
Keywords: Geometry of numbers; critical determinant; simultaneous Diophantine approximation 
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Nowak, Werner Georg. Estimating the critical determinants of a class of three-dimensional star bodies. Communications in Mathematics, Tome 25 (2017) no. 2, pp. 149-157. http://geodesic.mathdoc.fr/item/COMIM_2017__25_2_a4/