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Any increasing concave function $f:\mathbb R\to \mathbb R$ has a nonnegative derivative up to a countable set $N\subset \mathbb R$. Moreover $f$ is locally Lipschitz, thus locally absolutely continuou …

To complete Narutaka OZAWA's answer in comment by a concrete example as asked in the OP, here is a bounded linear functional on $c_0$ not attaining its norm w.r.to (an equivalent) strictly convex norm …

You are facing the classical optimal transport problem, on which there is a huge literature. Here is a recent comprehensive treatise by Cédric Villani (Warning: 1K pages).

I assume $a$ and $b$ are not linearly dependent: if they are, the objective takes a form $f(s)=\phi(a^Ts)$ and the problem reduces to a linear optimization. Writing the gradient of $f(s):={\sqrt …

I think there are simple counterexamples of the form $f(x,y)=\phi(x)+\psi(y)$ (here $x$ and $y$ denote real variables), where $\phi$ and $\psi$ are smooth, positive, decreasing, strictly convex funct …

For $0\le \theta\le\pi/2$, say that $T:H\to H$ is $\theta$-monotone (therefore monotone) iff for all $x$, $y$ in $H$, $(x-y,Tx-Ty)\ge \|x-y\|\,\|Tx-Ty\|\cos\theta$, that is, $x-y$ and $Tx-Ty$ make an …

This is the celebrated Chebyshev problem. The answer is positive in $\mathbb{R}^n$, and still open in the Hilbert space.