Example of geodesic quasiconvex function which is not geodesic convex.

Question

A geodesic quasiconvex function $f$ is a real valued function on complete Riemannian manifold $M$ such that for any points $x,y\in M$ and any geodesic $\gamma:[0,1]\rightarrow M$ with $\gamma(0)=x$ and $\gamma(1)=y$ the following relation holds $$f(\gamma(t))\leq {\rm max}\ \{f(x),f(y)\},\ \forall t\in[0,1].$$ Is there any example of geodesic quasiconvex function which is not geodesic convex in complete Riemannian manifold other than $\mathbb{R}^n$? Thank you.

Answer 1

Sure. It would be very strange if the implication quasiconvex $\implies$ convex were true on all complete Riemannian manifolds except the Euclidean space. For an explicit example, let $M=\mathbb{R}\times S^1$ and define $f:M\to\mathbb R$ by $f(x,\zeta) = \sqrt{|x|}$, where $x\in\mathbb{R}$ and $\zeta\in S^1$. Any geodesic curve on $M$ is of the form $\gamma (t) =(at+b, e^{cit})$, so $f\circ \gamma$ is quasiconvex but not convex.

(On the other hand: on some manifolds, like $S^n$, every quasiconvex function is constant and therefore convex.)