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The sharp Whitney extension theorem for convex $C^1$ Lipschitz functions

Carlos Mudarra

TL;DR

The paper addresses the problem of extending a $1$-jet $(f,G)$, defined on an arbitrary set $E\subset \mathbb{R}^n$, to a convex $C^1$ function on all of $\mathbb{R}^n$ with the sharp Lipschitz constant $\mathrm{Lip}(F) = \sup_{x\in E} |G(x)|$. It develops a decomposition theory for Lipschitz convex functions, constructs a sequence of minimal-extensions that preserve the bound on $|G|$, and uses an inf-convolution with the bound $L$ to produce a $C^1$ convex extension whose gradient matches the jet on $E$. The main contribution is establishing the sharp Lip constant in the unbounded-set setting and prescribing global behavior via a chosen subspace of directions of coercivity $X$, thereby extending the AM19APDE framework to the sharp convex Whitney setting. This advances the understanding of convex $C^1$ Whitney extensions and provides tools with potential applications to convex Lusin-type results and analysis in infinite-dimensional or unbounded contexts.

Abstract

For an arbitrary set $E \subset \mathbb{R}^n$, and functions $f:E \to \mathbb{R}$, $G: E\to \mathbb{R}^n$ with $G$ bounded, we construct $C^1(\mathbb{R}^n)$ convex extensions $(F, \nabla F)$ of $(f,G)$ with the sharp Lipschitz constant $$ \mathrm{Lip}(F) = \sup_{x\in E} |G(x)|, $$ provided that $(f,G)$ satisfies the pertinent necessary and sufficient conditions for $C^1$ convex, and Lipschitz extendability. Also, these extensions can be constructed with prescribed global behavior in terms of directions of coercivity.

The sharp Whitney extension theorem for convex $C^1$ Lipschitz functions

TL;DR

The paper addresses the problem of extending a -jet , defined on an arbitrary set , to a convex function on all of with the sharp Lipschitz constant . It develops a decomposition theory for Lipschitz convex functions, constructs a sequence of minimal-extensions that preserve the bound on , and uses an inf-convolution with the bound to produce a convex extension whose gradient matches the jet on . The main contribution is establishing the sharp Lip constant in the unbounded-set setting and prescribing global behavior via a chosen subspace of directions of coercivity , thereby extending the AM19APDE framework to the sharp convex Whitney setting. This advances the understanding of convex Whitney extensions and provides tools with potential applications to convex Lusin-type results and analysis in infinite-dimensional or unbounded contexts.

Abstract

For an arbitrary set , and functions , with bounded, we construct convex extensions of with the sharp Lipschitz constant provided that satisfies the pertinent necessary and sufficient conditions for convex, and Lipschitz extendability. Also, these extensions can be constructed with prescribed global behavior in terms of directions of coercivity.
Paper Structure (9 sections, 11 theorems, 102 equations)

This paper contains 9 sections, 11 theorems, 102 equations.

Table of Contents

  1. Introduction and main results
  2. Global behavior of Lipschitz convex functions
  3. Proof of Theorem \ref{['thm:maintheorem']}
  4. The first minimal function
  5. New data preserving the Lipschitz constant
  6. Extending to finitely-many points
  7. The second minimal function
  8. The first $C^1$ convex extension
  9. The sharp $C^1$ convex extension

Key Result

Theorem 1.1

Let $E \subset \mathbb{R}^n$ be a subset, $X\subset\mathbb{R}^n$ a linear subspace, and $f:E \to \mathbb{R}, \: G: E \to \mathbb{R}^n$ a $1$-jet with $G$ non-constant. Assume that $f,G,X$ satisfy conditions condition:convexity, condition:subspacecontained, condition:existencecones, condition:corners

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 12 more