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The paper http://arxiv.org/abs/1402.1561 might be of use in case $d = 2$. Your set $\mathcal{K}(\underline x)$ is denoted as $\mathrm{Conv}(X)$, see (3). A characterization of this set can be found i …

Set $G := \{(x,y) \in X \times Y : y = T(x)\}$. I think for $\gamma = \gamma_T$ we just have \begin{align*} \gamma(G) = \int_{X \times Y} \chi_G(x,y) \,\mathrm{d}\gamma(x,y) = \int_X \chi_G \mathbin\c …

If I understand your comment correctly, you are minimizing over a set $$ U := \{ u \in H^1(\Omega) \mid a \le u \le b \} $$ for some $a, b \in L^\infty(\Omega)$. Such a set is polyhedric in the sense …

If your problem has a solution $x^* \ne 0$, then $0$ is also a solution. Indeed, consider the function $$\varphi(t) = c(t \, x^*) - k\cdot (t \, x^*).$$ Since $x^*$ is a solution, we have $$\varphi(0) …

Let us assume that your $f$ is at least directionally differentiable at $x_0$ and that the directional derivative depends continuously on the direction (this may be satisfied under rather general assu …

I was also struggling with Exercise 2.36... I think that I am now able to solve it, although it seems that it is more difficult than it appears... The key seems to be the following theorem. Theorem: …