I am having some trouble understanding part of this paper.
Let $\psi$ be the unique solution of the optimization problem $\min \|\psi\|_2$ over all $\psi \in \{f : |f(x)-f(y)| \le L |x-y|^\beta\}$ with $\psi(0) \ge 1$. It is known that $\psi$ is an even function with compact support, and $\psi(0)=1>|\psi(x)|$ for $x \ne 0$.In the case $0<\beta \le 1$, one can easily show that $$\psi(x) = \mathbf{1}\{|x|\le 1\} (1-|x|^\beta).$$
Can someone explain how to justify these results? Intuitively, I see that you just want to go to zero as quickly as possible as $x$ goes away from zero while still respecting the smoothness condition, but I don't know how to formalize this.