The Hadwiger Conjecture for Unconditional Convex Bodies

We investigate the famous Hadwiger Conjecture, focusing on unconditional convex bodies. We establish some facts about covering with homotethic copies for Lp balls, and prove a property of outer normals to unconditional bodies. We also use a projection method to prove a relation between the bounded and unbounded versions of the conjecture. 1 Definitions and Background Definition 1. A convex body (closed and compact) in R is called unconditional if it is symmetric with respect to the coordinate hyperplanes. Definition 2. For convex bodies K and T , let N(K,T ) be the minimum number of translates of T needed to cover K. Conjecture 1 (Hadwiger). For K a convex body in R, N(K, int(K)) ≤ 2.