Volume bounds for shadow covering

TL;DR

This paper investigates volume inequalities for convex sets based on their shadow covering properties, establishing bounds and conditions under which one set's volume exceeds or is bounded by another's.

Contribution

It provides new bounds on volumes of convex sets based on projection and shadow covering conditions, including a universal constant bound independent of dimension.

Findings

Constructed examples where volume of K exceeds L despite shadow covering

Established that (n/(n-1))L contains a translate of K under shadow covering

Derived a universal volume bound V_n(K) <= 2.942 V_n(L)

Abstract

For n >= 2 a construction is given for a large family of compact convex sets K and L in n-dimensional Euclidean space such that the orthogonal projection L_u onto the subspace u^\perp contains a translate of the corresponding projection K_u for every direction u, while the volumes of K and L satisfy V_n(K) > V_n(L). It is subsequently shown that, if the orthogonal projection L_u onto the subspace u^\perp contains a translate of K_u for every direction u, then the set (n/(n-1))L contains a translate of K. If follows that V_n(K) <= (n/(n-1))^n V_n(L). In particular, we derive a universal constant bound V_n(K) <= 2.942 V_n(L), independent of the dimension n of the ambient space. Related results are obtained for projections onto subspaces of some fixed intermediate co-dimension. Open questions and conjectures are also posed.