Normality and modulability indices. Part II: Convex cones in Hilbert spaces

Let K be a closed convex cone in a Hilbert space X. Let BX be the closed unit ball of X and K• = (BX +K) ∩ (BX −K). The normality index ν(K) = sup{r 0: rK• ⊂ BX} is a coefficient that measures to which extent the cone K is normal. We establish a formula that relates ν(K) to the maximal angle of K. A concept dual to normality is that of modulability. As a by-product one obtains a formula for computing the modulability index μ(K) = sup r 0: rBX ⊂ K• of K. The symbol K• stands for the absolutely convex hull of K ∩ BX. We show that μ(K) can be expressed in terms of the smallest critical angle of K. © 2007 Elsevier Inc. All rights reserved