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

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 $(F,\mathrm{\nabla }F)$ of $(f,G)$ with the sharp Lipschitz constant $$\mathrm{L}\mathrm{i}\mathrm{p}(F)=\underset{x\in E}{\sup }\mathrm{\mid }G(x)\mathrm{\mid },$$ 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.