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1 Basic Properties in Euclidean Spaces Convex Functions: Theory, Examples and Applications Contents with Comments/Questions 1. Start to build interface/database for convex functions. I. BASIC THEORY 1 Basic properties in Euclidean Spaces 1.1 Basic Continuity & Differentiability properties Definition, sublinearity implies positively homogeneous. One-sided derivatives, sublinearity of right- hand derivative. Three slope inequality for convex functions on R; increasing derivative for convex n functions on R, 2nd derivative nonnegative. Hessian matrix for convex functions on R . Continuity, Lipschitzness and local boundedness properties. Convex function on simplex implies continuity on interior of simplex. Separation/Hahn Banach theorems will be stated in various forms with proofs relegated to references and guided exercises. Gateaux and Fr´echet derivatives. 2 Example 1.1. There exists f : R → R such that f is Gateaux differentiable everywhere, but f is not continuous at (0, 0). x4y ∞ Proof. Let f(x, y) = x8+y2 if (x, y) 6= (0, 0) and let f(x, y) = (0, 0) otherwise. Then f is C -smooth if (x, y) 6= (0, 0), one can check that the Gateaux derivative of f is 0 at (0, 0), but f is not continuous at (0, 0). Fr´echet differentiability implies continuity for general functions. Gateaux differentiability implies Fr´echet differentiability for locally Lipschitz functions. If f is convex and Gateaux differentiable at n n x0 ∈ R , then f is continuous and Fr´echet differentiable at x0. If f : R → R is convex and ∂f(x0) is a singleton, then f is Fr´echet differentiable at x0. 1.2 Conjugate properties and duality • A tighter but trimmed down version of infinite dimensional case that is currently under construction. 1.3 Differentiability (generic, porous & measure theoretic) • Will be built after the infinite dimensional chapter. 1.4 Semi-definite matrices and eigenvalue functions • Borwein-Lewis and Borwein-Zhu have some hints. Zipped versions follow 1.5 Support and extremal structure • Borwein-Lewis have a good treatment of Krein-Mil’man, and of polyhedral issues. • Material on exposing and strongly exposing faces of sets and cones needs to be added (Borwein- Wolkowicz and more recent semi-definite work). • Zalinescu’s recent book may help. I’ll send a PDF copy. 1.6 Higher order properties • Proof of Alexandrov’s theorem, coincidence of various second order notions (see Borwein-Noll) 2 Basic properties in Hilbert and Banach spaces 2.1 Basic Continuity and Differentiability properties • Local boundedness above and Lipschitzian properties Proposition 2.1. A lsc convex function is continuous at all points of the interior of its domain. Note, lsc is needed in infinite dimensions. For example discontinuous linear functional. • definitions of different types of differentiability; symmetric definitions of differentiability; one-sided derivatives. Gateaux Differentiability Proposition 2.2. Suppose f is lsc convex and Gateaux differentiable at x, the f is continuous at x. Example 2.3. Let X be a separable Banach space, then there is an lsc convex function on X such that ∂f(0) is a singleton, but f is not Gateaux differentiable at 0. Note to JMB, what is the proof for this in general nonseparable Banach spaces as alluded to in the Borwein-Lewis book? Got me! I think an example exists iff there is a closed densely spanning convex set without interior, hence in WCG spaces, spaces with such quasi-complements etc Example 2.4. On any infinite dimensional Banach space there is a convex function that is Gateaux differentiable at 0, lsc at 0, but discontinuous at 0. Proof. Let φ be a discontinuous linear functional on X. Define f(x) = φ2(x). Then f is discontinuous, f is Gateaux differentiable at 0 with f 0(0) = 0. The following for β-differentiability: • Smulyan’sˇ criterion From Smulyan’s theorem can have definition/exercises concerning various topological upper semicon- tinuity notions of ∂f as a set valued map related to appropriate differentiability. Explicitly state continuity properties for the Gateaux/Frechet differentiability cases. • implicit derivatives for gauges created from level sets of convex functions 2.2 Conjugate properties and duality • definitions and characterizations of various types of extreme points for convex functions; do this very carefully and clearly. Will be useful later for compare and contrast with essential smoothness and essential strict convexity. • Fenchel conjugates; bi-conjugates. H¨ormander’s theorem Conjugates of infimal convolutions; duality between extreme points and smooth points, between continuity and compactness (Moreau). • Strictly convex, locally uniformly convex and uniformly convex functions, duality with smoothness. β-versions can be stated as exercises. Differentiability properties of distance functions when dual norm has appropriate rotundity properties. At least state and outline proofs as guided exercises via various approaches (i.e. conjugates as infimal convolutions, via Smulyan’s criterion directly as can be found in DGZ). Refer perhaps forward to examples where smoothness of norm alone doesn’t imply smoothness of distance function. In this section, we won’t prove or even state existence results for certain norms. We may simply suggest the definitions (a norm is locally uniformly convex if f(x) = kxk2 is a locally uniformly convex function (as we’ve defined), then say i.e.... and leave it at that). 2.3 Differentiability theory (β-differentiability, generic, porous & measure theo- retic) • Sizes of sets of differentiability: see e.g. Phelps; Benyamini and Lindenstrauss. This is a place to point to the literature rather than tell and prove the complete picture. We should paraphrase Benyamini and Lindenstrauss, and add some nice examples. Warren Moors can assist. 2.4 Renorming theory, an introduction • smooth and rotund norms; duality, types of spaces and survey of key existence results (classical spaces, wcg/reflexive spaces). Existence of rotund/smooth norms and coercive convex functions with same smoothness property. Notes on C(K) spaces, especially, when K scattered. A few proofs in separable spaces should be relatively completely sketched Note: we will have developed most of the tools to outline how uniformly (Gateaux) smooth norms can be constructed from uniformly (Gateaux) smooth bumps as perhaps guided exercises with references to appropriate literature such as papers by Fabian-Zizler, McLaughlin-V, Tang. 2.5 Integral functionals • Sketch basic theory of normal convex integrands (Rockafellar, Ioffe, Balder). Main conditions ensuring that ∗ ∗ If = (If ) R and consequences for ∂If , where If (x) = T f(x(t), t) µ(dt). 2.6 Support structure • Relation between RNP, Asplund and Krein-Milman properties and how they are expressed for convex functions. James, theorem and consequences. 2.7 Weak vs norm convergence ∗ • LUR, Kadec-Klee, Borwein-Lewis L1 results (If is strongly rotund iff f is smooth and everywhere defined) and Theorem 2.5. Suppose fn(t) ∈ Ω(t) a.e and fn + f ∈ extΩ(t) a.e. Then fn →L1 f. 2.8 Monotone operators and convex functions • begin with material in B-Z and B-L. State Mignot’s result, see B-Noll. JMB to add new results later. 2.9 Higher order properties • Update of material in Borwein-Noll, failure of Alexandrov except possibly in separable `2. • Outline such as Azagra & Ferrera, and their, or a similar, approach to C∞-smooth approximations (finite dimensions) but I’m not sure what is core. State Deville/Fonf/Hajek; Cepadello/Hajek and Fry approximation results, and Frontisi’s result that convex approximation implies smooth partitions. II. MORE ADVANCED THEMATIC CHAPTERS 3 Essential Smoothness and Essential Strict Convexity in Banach space • Ask Heinz for his latest work here. Two crucial things to accomplish here. Have enough for examples and applications (we may refer forward to Part III). Highlight the niceness/tightness of duality in contrast to the failure of such for smoothness and strict convexity. 4 Dual Topologies in Banach Spaces and Properties of Convex Functions • Write survey paper to get idea for contents of this chapter. This should be our next task after signing off on this outline. 5 Approximation Results and Inverse Problems • The relevant sections in Borwein-Zhu and Borwein-Lewis are a good starting point. See also Boyd’s new book on Convex Programming (it is on the web). III. APPLICATIONS AND EXAMPLES Fields from which examples will be categorized include: 1. Statistics and Design 2. Optimization and Approximation 3. Engineering and Control 4. Economics and Information Theory 5. Classical Analysis 6. Inequalities and Functional Equations 7. Discrete Mathematics and Combinatorics 8. Computer Science and Information Technology 9. Other: Social, Physical, Biological Sciences 2. Euclidean space constructions 1. Geometric characterizations 2. Barrier Functions and Entropies 3. Spectral and Eigenvalue Functions 4. Bregman/Legendre Functions 5. Symbolic and Numeric Convex Analysis 6. Other 3. Banach space constructions 1. Geometric characterizations 2. Barrier Functions and Entropies 3. Integral Functionals 4. Legendre Functions 5. Other.
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