Total Convexity for Powers of the Norm in Uniformly Convex Banach Spaces
نویسندگان
چکیده
The function Df , called the Bregman distance associated with f , is always well defined because ∂f(x) is nonempty and bounded, for all x ∈ X (see, e.g., [22]), so that the infimum in (1.1) cannot be −∞. It is easy to check that Df (x, y) ≥ 0 and that Df (x, x) = 0, for all x, y ∈ X. If f is strictly convex then Df (x, y) = 0 only when x = y. For t ∈ [0,∞) and z ∈ X let U(z, t) = {x ∈ X : ‖x− z‖ = t}. Following [5], we define νf : X × [0,∞) → [0,∞) as νf (z, t) = inf {Df (x, z) : x ∈ U(z, t)}. (1.2)
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