# Covering minima and lattice point free convex bodies

## Abstract

Suppose *K* is a convex set of nonzero volume in Euclidean *n*-space *R*^{n} and it is symmetric about the origin (i.e., if *x* belongs to *K*, so does - *x*). For any real number *t*, let *tK*={*tx* : *x*∈*K*}. The infimum over all positive real numbers *t* such that if a copy of *tK* is placed centered at every integer point, all of *R*^{n} is covered, is called the “covering radius” of *K* (with respect to the lattice *Z*^{n}). The covering radius and related quantities have been studied extensively in Geometry of Numbers. In this paper, we define and study the “covering minima” of a convex body which is not necessarily symmetric about the origin; the covering radius will be a special case of one of of these minima. This extension to general convex bodies has among other things, applications to algorithms for Integer Programming which was our initial motivation. This motivation is explained in some detail later. We use the results of the paper to derive bounds on the width of lattice point free convex bodies and analyze their structure.

## Keywords

Distance Function Lattice Point Convex Body Polynomial Time Algorithm Covering Minimum## Preview

Unable to display preview. Download preview PDF.

## References

- J. Bourgain and V.D. Milman,
*Sections euclidiennes et volume des corps symetriques convexes dans R*^{n}, C.R.Acad. Sc. Paris, t. 300, Série I,n 13, (1985) pp435–438Google Scholar - J.W.S.Cassels,
*An introduction to the geometry of numbers*Springer Verlag (1971)Google Scholar - A.Frank and E.Tardos,
*An application of simultaneous approximation in combinatorial optimization*, Report Institut für Ökonometrie und Operations Research, Uni. Bonn, W.Germany (1985) to appear in Combinatorica.Google Scholar - J.Hastad,
*Dual Witnesses*Manuscript (1986)Google Scholar - J.Hastad, private communication (1986a)Google Scholar
- R.Kannan,
*Improved algorithms for integer programming and related lattice problems*15 th Annual ACM symposium on theory of computing (1983) pp193–206. Revised version*Minkowski's Convex body theorem and integer programming*, Carnegie-Mellon University Computer Science Dept. Technical Report CMU-CS-86-105 (1986)Google Scholar - N. Karmarkar,
*A new polynomial time algorithm for linear programming*, Combinatorica 4, pp373–396 (1984)Google Scholar - A. Korkine and G. Zolotarav,
*Sur les formes quadratiques*, Math. Annalen 6, (1873) pp 366–389Google Scholar - J.Lagarias, H.W.Lenstra and C.P.Schnorr,
*Korkine-Zolotarev bases and successive minima of a lattice and its reciprocal lattice*, Manuscript (1986)Google Scholar - C.G. Lekkerkerker,
*Geometry of Numbers*North Holland, Amsterdam, (1969)Google Scholar - A.K. Lenstra, H.W. Lenstra and L. Lovász,
*Factoring polynomials with rational coefficients*Mathematische Annalen 261 (1982), pp513–534Google Scholar - H.W. Lenstra,
*Integer programming with a fixed number of variables*First announcement (1979) Mathematics of Operations research, Volume 8, Number 4 Nov (1983) pp 538–548Google Scholar - L. Lovász,
*An algorithmic theory of numbers, graphs and convexity*, Report number 85368-OR, Institut für Operations Research, univerität Bonn, Bonn (1985)Google Scholar - K.Mahler,
*On lattice points in polar reciprocal domains*, Proc. Kon. Ned. Wet. 51 pp 482–485 (=Indag. Math. 10, pp176–179) (1948)Google Scholar - J. Milnor and D. Husemoller,
*Symmetric bilinear forms*Springer-Verlag, Berlin (1973).Google Scholar