On hereditarily dentable sets in Banach spaces

The note deals with closed convex bounded hereditarily dentable sets in Banach spaces. As an example let us cite the following result: a closed convex bounded set $B$ is hereditarily dentable iff it is hereditarily $f$-dentable (i.e. $\forall K\subset B$, $\forall\varepsilon>0$, $\exists z\in K$: $z\not\in\mathrm{co} (K\setminus\{x\|x-z\|\le\varepsilon\}))$ and iff each closed subset of $B$ has an extreme point. The proof of the first equivalence (which is the main theorem of the paper) is based only on the definition of dentability and differs essen-tially from the Davis–Phelps proof for the special case $B=\{x:\|x\|\le1\}$.