Upper and lower bounds for the Bregman divergence.

In this paper we study upper and lower bounds on the Bregman divergence Δ F ξ ( y , x ) : = F ( y ) - F ( x ) - 〈 ξ , y - x 〉 for some convex functional F on a normed space X , with subgradient ξ ∈ ∂ F ( x ) . We give a considerably simpler new proof of the inequalities by Xu and Roach for the special case F ( x ) = ∥ x ∥ p , p > 1 . The results can be transferred to more general functions as well.