Covering convex bodies by cylinders and lattice points by flats

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.

References in corpus (3)

• The plank problem for symmetric bodies
• Covering lattice points by subspaces
• Distances between non--symmetric convex bodies and the $M{M}^{\ast }$-estimate

Cited by in corpus (4)

• Bang's problem and symplectic invariants
• Restricted Successive Minima
• Packing convex bodies by cylinders
• Minkowski's successive minima in convex and discrete geometry