Recently, Venkatesh improved the best known lower bound for lattice sphere
packings by a factor $\log\log n$ for infinitely many dimensions $n$. Here we
prove an effective version of this result, in the sense that we exhibit, for
the same set of dimensions, finite families of lattices containing a lattice
reaching this bound. Our construction uses codes over cyclotomic fields, lifted
to lattices via Construction A
The classes of sums of arithmetic-geometric exponentials (SAGE) and of sums of nonnegative circuit polynomials (SONC) provide nonnegativity certificates which are based on the inequality of the arithmetic and geometric means. We study the cones of symmetric SAGE and SONC forms and their relations to the underlying symmetric nonnegative cone.As main results, we provide several symmetric cases where the SAGE or SONC property coincides with nonnegativity and we present quantitative results on the differences in various situations. The results rely on characterizations of the zeroes and the minimizers for symmetric SAGE and SONC forms, which we develop. Finally, we also study symmetric monomial mean inequalities and apply SONC certificates to establish a generalized version of Muirhead's inequality.
The maximal density of a measurable subset of R n avoiding Euclidean distance 1 is unknown except in the trivial case of dimension 1. In this paper, we consider the case of a distance associated to a polytope that tiles space, where it is likely that the sets avoiding distance 1 are of maximal density 2 −n , as conjectured by Bachoc and Robins. We prove that this is true for n = 2, and for the Voronoï regions of the lattices An, n ≥ 2.
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