7 - Barriers and Legendre functions

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 : ℝ^N → (-∞,+∞] is:

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