Fixed points and boundaries

A lemma of Ludvik Janos is used to show that if a nonexpansive self-map $T$ of a compact set $X$ is contractive on $\Delta 'X$, the boundary of $X$ in $\overline {{\text{co}}} X$, then $T$ has a fixed point in $X$. It is further proven that if $T(\Delta 'X) \cap \Delta 'X = \emptyset$, or if $T$ maps any point $y$ of $X$ away from $\Delta 'X$, then $T$ has a fixed point in $X$.