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7 - Barriers and Legendre functions

Published online by Cambridge University Press:  07 September 2011

Jonathan M. Borwein
Affiliation:
University of Newcastle, New South Wales
Jon D. Vanderwerff
Affiliation:
La Sierra University, California
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Summary

Considerable obstacles generally present themselves to the beginner, in studying the elements of Solid Geometry, from the practice which has hitherto uniformly prevailed in this country, of never submitting to the eye of the student, the figures on whose properties he is reasoning, but of drawing perspective representations of them upon a plane. … I hope that I shall never be obliged to have recourse to a perspective drawing of any figure whose parts are not in the same plane.

(Augustus De Morgan)

Essential smoothness and essential strict convexity

This chapter is dedicated to the study of convex functions whose smoothness and curvature properties are preserved by Fenchel conjugation.

Definition 7.1.1. We will say a proper convex lower-semicontinuous function f : ℝN → (-∞,+∞] is:

  1. (a) essentially smooth in the classical sense if it is differentiable on int dom f ≠, and ∥∇f(xn)∥ → ∞ whenever xnx ∈ bdry dom f;

  2. (b) essentially strictly convex in the classical sense, if it is strictly convex on every convex subset of dom ∂f;

  3. (c) Legendre in the classical sense, if it is both essentially smooth and essentially strictly convex in the classical sense.

The duality theory for these classical functions is presented in [369, Section 26]. The qualification in the classical sense is used to distinguish the definition on ℝn from the alternate definition for general Banach spaces given below which allows the classical duality results to extend to reflexive Banach spaces as shown in [35], which we follow.

Type
Chapter
Information
Convex Functions
Constructions, Characterizations and Counterexamples
, pp. 338 - 376
Publisher: Cambridge University Press
Print publication year: 2010

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