Differentiable convex extensions with sharp Lipschitz constants

Given a superreflexive Banach space $X$, and a set $E \subset X$, we characterise the $1$-jets $(f,G)$ on $E$ that admit $C^{1,\omega}$ convex extensions $(F,DF)$ to all of $X$; where $\omega$ is any admissible modulus of continuity depending on the regularity of $X$. Moreover, we obtain precise estimates for the growth of the $C^{1,\omega}$ seminorm of the extensions with respect to the initial data. We show how these estimates can be improved in the Hilbert setting, and are asymptotically sharp for H\"older moduli. Remarkably, our extensions have the sharp Lipschitz constant $\mathrm{Lip}(F,X) = \|G\|_{L^\infty(E)}$, when $G$ is a bounded map. All these extensions are given by simple and explicit formulas. We also prove a similar theorem for $C^1$ convex extensions of jets defined on compact subsets $E$ of superreflexive spaces $X$, with the sharp Lipschitz constant too. The results are new even when $X=\mathbb{R}^n.$