AN INCLUSION–EXCLUSION IDENTITY FOR NORMAL CONES OF POLYHEDRAL SETS. Issue 1 (5th February 2018) - Hug, Daniel; Kabluchko, Zakhar
Abstract : For a non‐empty polyhedral set P ⊂ R d, let F ( P ) denote the set of faces of P, and let N ( P, F ) be the normal cone of P at the non‐empty face F ∊ F ( P ) . We prove the identity ∑ F ∈ F ( P ) ⁡ ( − 1 ) dim ⁡ F 𝟙 F − N ( P, F ) = 1 if P is bounded, 0 if P is unbounded and line‐free . Previously, this formula was known to hold everywhere outside some exceptional set of Lebesgue measure 0 or for polyhedral cones. The case of a not necessarily line‐free polyhedral set is also covered by our general theorem.