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In this paper the simple structure between some convex sets in the Banach spaceH introduced by Hornich is used to determine the extreme points of the familiesK(α) of convex functions of order α andV(k) of functions with bounded boundary rotationkπ. For close-to-convex functions of order β,β∈]0,1[, a partial result is given. The results forK(α) andV(k) agree with those that hold for the closed convex hulls of the same families with respect to the usual linear structure and the topology of locally uniform convergence. However, in this case, fork∈]2,4[ the question of determining the extreme points of<img src="/fulltext-image.asp?format=htmlnonpaginated&src=X4M489720W772480_html\605_2005_Article_BF01295667_TeX2GIFIE1.gif" border="0" alt="$$\overline {co} $$" />V(k) is still open.

Classical families of univalent functions in the Hornich space