Abstract
Every convex body K in R n has a coordinate projection PK that contains at least vol ( 1 6 K ) cells of the integer lattice P Z n , provided this volume is at least one. Our proof of this counterpart of Minkowski's theorem is based on an extension of the combinatorial density theorem of Sauer, Shelah and Vapnik–Chervonenkis to Z n . This leads to a new approach to sections of convex bodies. In particular, fundamental results of the asymptotic convex geometry such as the Volume Ratio Theorem and Milman's duality of the diameters admit natural versions for coordinate sections.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.