Covering convex bodies by cylinders and lattice points by flats

TL;DR

This paper establishes lower bounds on the sum of cylinder base volumes covering convex bodies and on the number of flats needed to cover integer points within these bodies, advancing understanding of geometric covering problems.

Contribution

It provides new lower bounds related to covering convex bodies with cylinders and covering lattice points with flats, addressing an open problem of Bang (1951).

Findings

Lower bounds for cylinder base volumes covering convex bodies.

Lower bounds on flats needed to cover lattice points.

Connections to an open problem of Bang (1951).

Abstract

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points of a given convex body in d-dimensional Euclidean space for 0<k<d.