I would like to know if there is a standard name for functions $f:[0,1]\to\mathbb R$ with the following convexity property:
$$ \forall s<t<u\qquad f(t)\leq\max\{f(s),f(u)\}$$
(the fact that the domain is $[0,1]$ and the codomain is $\mathbb R$ is not the important thing)
Motivation.
Consider a real vector space $X$ and ist projective $\mathbb P(X)=X/\sim$ where $x\in y$ iff there is $\lambda\neq 0$ such that $x=\lambda y$.
Now consider a function $f:X\to \mathbb R$ which is scale invariant, hence projects to a function on $\mathbb P(X)$. For $p,q\in X$ the segment between $p$ and $q$ is parametrized by $$tq+(1-t)p$$
The segment $\overline{pq}$ is projectively well-defined.
Unfortunately, the parametrization is not scale invariant (the middle point of $\overline{pq}$ is not equivalent to the middle point of $\overline{p (100q)}$) thus the convexity of a function is not a projetive invariant.
However, if $f$ is convex for some parametrization of the segment between $p$ and $q$ then it is true that $f(x)\leq \max\{f(p),f(q)\}$ for any $x$ in the segment. On the other hand, if a function has that property for any subsegment of $\overline{pq}$, then it has no local maxima and there is a reparametrization of $\overline{pq}$ such that $f$ becomes convex in the usual sense.
As pointed out in comments, the property that characterizes functions such that $$\forall s<t<u\qquad f(t)\leq\max\{f(s),f(u)\}$$
is Quasiconvexity. In particular a function satisfying the above inequality is called quasiconvex (or quasi-convex or quasi convex) and it is called strict quasiconvex if the inequality is strict.
See article on wikipedia.