Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

4 Issues per year


IMPACT FACTOR 2017: 0.734

CiteScore 2017: 0.70

SCImago Journal Rank (SJR) 2017: 0.695
Source Normalized Impact per Paper (SNIP) 2017: 0.891

Mathematical Citation Quotient (MCQ) 2017: 0.62

Online
ISSN
1615-7168
See all formats and pricing
More options …
Ahead of print

Issues

A proof of a conjecture by Haviv, Lyubashevsky and Regev on the second moment of a lattice Voronoi cell

Alexander Magazinov
Published Online: 2018-07-20 | DOI: https://doi.org/10.1515/advgeom-2018-0018

Abstract

In this short note we prove a sharp lower bound for the second moment of a lattice Voronoi cell in terms of the respective covering radius. This gives an affirmative answer to a conjecture by Haviv, Lyubashevsky and Regev. We also characterize those lattice Voronoi cells for which this lower bound is attained.

Keywords: Voronoi cell; lattice; covering radius

MSC 2010: 11H31

References

  • [1]

    S. Artstein-Avidan, A. Giannopoulos, V. D. Milman, Asymptotic geometric analysis. Part I, volume 202 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2015. MR3331351 Zbl 1337.52001Google Scholar

  • [2]

    L. Danzer, B. Grünbaum, Über zwei Probleme bezüglich konvexer Körper von P. Erdős und von V. L. Klee. Math. Z. 79 (1962), 95–99. MR0138040 Zbl 0188.27602CrossrefGoogle Scholar

  • [3]

    M. M. Deza, M. Laurent, Geometry of cuts and metrics, volume 15 of Algorithms and Combinatorics. Springer 2010. MR2841334 Zbl 1210.52001Google Scholar

  • [4]

    I. Haviv, V. Lyubashevsky, O. Regev, A note on the distribution of the distance from a lattice. Discrete Comput. Geom. 41 (2009), 162–176. MR2470075 Zbl 1163.68040CrossrefWeb of ScienceGoogle Scholar

  • [5]

    M. W. Meckes, Sylvester’s problem for symmetric convex bodies and related problems. Monatsh. Math. 145 (2005), 307–319. MR2162349 Zbl 1081.52008CrossrefGoogle Scholar

  • [6]

    O. Regev, N. Stephens-Davidowitz, A reverse Minkowski theorem. In: STOC’17–Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, 941–953, ACM, New York 2017. MR3678241 Zbl 1370.11073Google Scholar

  • [7]

    U. Shapira, B. Weiss, Stable lattices and the diagonal group. J. Eur. Math. Soc. 18 (2016), 1753–1767. MR3519540 Zbl 1370.11074Web of ScienceCrossrefGoogle Scholar

  • [8]

    A. C. Woods, On a theorem of Tschebotareff. Duke Math. J. 25 (1958), 631–637. MR0132139 Zbl 0084.04602CrossrefGoogle Scholar

About the article


Received: 2017-10-26

Revised: 2018-03-14

Published Online: 2018-07-20


Funding: Supported in part by ERC Starting Grant 678520.


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0018.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in