Let $\Omega$ be a convex domain in $\mathbb{R}^{n}$, then

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  $u_{h}$ is linear in a $v$-direction in $\Omega$ if $u$ is linear in the $v$-direction in the Minkowski sum $\Omega+h(-P)$, with $\langle v,\nabla u_{h}\rangle=\langle v,\nabla u\rangle$.

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  $u_{h}$ is convex in $\mathbb{R}^{n}$ if $u$ is. The gradient $\nabla u_{h}(x),x\in\Omega$, is always inside the convex hull of all possible gradients of $u$ in $\Omega+h(-P)$.

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  $u_{h}$ is strictly convex in $\Omega+h(-P)$ if $u$ is convex in $\mathbb{R}^{n}$ and strictly convex in $\Omega$.