For differentiable function $f:\mathbb R\to\mathbb R$, saying that $f$ has only one same local extrema $f(x)$ is equivalent of saying $f$ is quasiconvex (concave). But for higher dimensional domains this statement is obviously not true, as quasiconvexity (concavity) is a much stronger statement.
My question is, what kind of convexity is equivalent of saying $f:\mathbb R^n\to\mathbb R$ has only one local extrema point? It is not "semi" or "pseudo" convexity (concavity).