Abstract
Let ‖ · ‖ be the euclidean norm on R n and let γn be the (standard) Gaussian measure on R n with density \((2\pi )^{ - n/2} e^{ - \left\| x \right\|^2 /2} \). Let ϑ (≃ 1.3489795) be defined by \(\gamma _1 ([ - \vartheta /2, \vartheta /2]) = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \) and let L be a lattice in R n generated by vectors of norm ≤ ϑ. Then, for any closed convex set V in R n with \(\gamma _n (V) \ge \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \), we have L + V = R n (equivalently, for any a ε R n, (a + L) ∩ V ≠ ∅). The above statement can also be viewed as a “nonsymmetric” version of the Minkowski theorem.
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Part of this research was done while the first author was visiting Case Western Reserve University under a cooperation grant from KBN (Poland) and NSF (USA). The second author was supported in part by the National Science Foundation.
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Banaszczyk, W., Szarek, S.J. Lattice coverings and Gaussian measures of n-dimensional convex bodies. Discrete Comput Geom 17, 283–286 (1997). https://doi.org/10.1007/PL00009294
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DOI: https://doi.org/10.1007/PL00009294