Almost everywhere convex functions on \(\mathbb R^n\) and weak derivatives (Q1864681)

In this paper the authors study characterizations of almost everywhere (a.e.) convex functions, that is functions that are equal a.e. to a convex function, defined on an open, bounded and convex subset \(\Omega\) of \(\mathbb R^n\). Their main results provide necessary or sufficient conditions for a function to be a.e. convex. In particular, they show that any function a.e. convex belongs to \(W^{1,p}_{\text{loc}}\) for every \(p\geq 1\) and satisfies the first order condition \(f(x)-f(y)\geq\langle\nabla f(y),x-y\rangle\) for almost all \(x,y\in\Omega\), where \(\nabla f\) denotes the weak gradient of \(f\). Conversely, they prove that if \(f\) belongs to \(W^{1,p^*}_{\text{loc}}(\Omega)\) for some \(p^*\geq 1\), and satisfies the first order condition a.e., then it is a.e. convex.