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Here is a counterexample to the convergence of $\nabla F$ in the conditions you give. Let $f(x)=\sqrt{1+x^2}$, let $s(x)=\cos(\ln(\ln(x^2+10)))$ and let $\phi:\mathbb{R}\to[0,1]$ be some $C^\infty$ bu …

If indeed $O\subseteq A$, we can still define $O'=O\cup(A\setminus K)$ and the rest of the proof works the same way (when proving convexity, we can define $r$ as the point of $[p,q)\cap\partial K$ closest …

Here is a more concrete construction. Let $X=\left\{\frac{n}{2^m};n\in\mathbb{Z},m\in\mathbb{N}\right\}\subseteq\mathbb{Q}$ and consider the set $$S=(\mathbb{R}\setminus\mathbb{Q})^2\cup\{(x,y)\in\mat …

It seems an path-connected anti-convex subset of $\mathbb{R}^2$ containing $(\mathbb{R}\setminus\mathbb{Q})^2$ exists. Firstly, let $A$ be a countable, dense subset of $\mathbb{R}^2$, and let $B$ be t …

Such a function doesn't exist for some choices of $a$. For notational purposes I will change the unit interval by $[-2,2]$. Consider a $C^\infty$ function $a:[-2,2]\to\mathbb{R}$ such that $a(0)=3$, $ …