An Optimization Problem Related to Minkowski’s Successive Minima

  • Authors:

  • Romanos Malikiosis

  • Affiliations:

  • UCLA, Department of Mathematics, Box 951555, 90095-1555, Los Angeles, CA, USA

  • Venue:
  • Discrete & Computational Geometry
  • Year:
  • 2010

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Abstract

The purpose of this paper is to establish an inequality connecting the lattice point enumerator of a 0-symmetric convex body with its successive minima. To this end, we introduce an optimization problem whose solution refines former methods, thus producing a better upper bound. In particular, we show that an analogue of Minkowski’s second theorem on successive minima with the volume replaced by lattice point enumerator is true up to an exponential factor, whose base is approximately 1.64.