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

Let K be a closed convex cone in a Hilbert space X  . Let BXBX be the closed unit ball of X   and K•=(BX+K)∩(BX−K)K•=(BX+K)∩(BX−K). The normality indexν(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)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 indexμ(K)=sup{r⩾0:rBX⊂K•} of K  . The symbol K•K• stands for the absolutely convex hull of K∩BXK∩BX. We show that μ(K)μ(K) can be expressed in terms of the smallest critical angle of K.