Convex function on nonconvex set and global minimum

Question

Can someone give an example of a convex function $f$ on a path-connected compact nonconvex set where some point $c$ is a local minimum with $\nabla f(c)=0$ but not a global minimum. Thus showing that the set being not convex makes finding global minimums harder.

Answer 1

Consider the set $$A:=[-1,3]\times[-3,3]\ \setminus\ \{(x,y)\ |\ 0<|y|<x\leq3\}$$ (a rectangle minus an isosceles triangle) and define $$f(x,y):=\cases{(x-2)^2 & $(x\leq 0 \ \vee\ y<0)$ \cr 2(x-1)^2+2 & $(x>0\ \wedge \ y>0)$\cr}\ .$$ This $f$ is $C^1$ on $A$, convex, and has local minima at $c:=(2,-{5\over2})$, $c':=(1,2)$ with $f(c)=0$, $f(c')=2$.

Answer 2

For example the function $f(x,y) = x^2+y^2$ restricted to the square with vertices $(-1,-2)$, $(5,-2)$, $(5,4)$ and $(-1,4)$ has four local minima but only one global minimum.