# Locally $C^{1,1}$ convex extensions of $1$-jets

## TL;DR

This paper characterizes when a 1-jet defined on an arbitrary subset of Euclidean space can be extended to a locally $C^{1,1}$ convex function, providing explicit formulas and applications to convex hypersurfaces.

## Contribution

It establishes necessary and sufficient conditions for such convex $C^{1,1}_{loc}$ extensions, along with explicit formulas and variants for broader regularity classes.

## Key findings

- Provided explicit formulas for convex $C^{1,1}_{loc}$ extensions.
- Characterized conditions for the existence of convex extensions of 1-jets.
- Applied results to construct convex hypersurfaces with prescribed tangent hyperplanes.

## Abstract

Let $E$ be an arbitrary subset of $\mathbb{R}^n$, and $f:E\to\mathbb{R}$, $G:E\to\mathbb{R}^n$ be given functions. We provide necessary and sufficient conditions for the existence of a convex function $F\in C^{1,1}_{\textrm{loc}}(\mathbb{R}^n)$ such that $F=f$ and $\nabla F=G$ on $E$. We give a useful explicit formula for such an extension $F$, and a variant of our main result for the class $C^{1, \omega}_{\textrm{loc}}$, where $\omega$ is a modulus of continuity. We also present two applications of these results, concerning how to find $C^{1,1}_{\textrm{loc}}$ convex hypersurfaces with prescribed tangent hyperplanes on a given subset of $\mathbb{R}^n$, and some explicit formulas for (not necessarily convex) $C^{1,1}_{\textrm{loc}}$ extensions of $1$-jets.