Mazur spaces and 4.3-intersection property of $(BM)$-spaces

The paper puts forward some combinatorial and geometric properties of finite-dimensional $(BM)$-spaces. A remarkable property of such spaces is that in these spaces one succeeds in giving an answer to some long-standing problems of geometric approximation theory, and in particular, to the question on the existence of continuous $\varepsilon$-selections on suns (Kolmogorov sets) for all $\varepsilon>0$. A finite-dimensional polyhedral $(BM)$-space is shown to be a Mazur space, satisfies the 4.3-intersection property, and its unit ball is proved to be a generating set (in the sense of Polovinkin, Balashov, and Ivanov).