Boundaries of a convex set and interpolation sets

Abstract

The structure of the "boundary" of a convex set C is a field of intensive research in functional analysis. The classical "boundary" is the set Ext(C) of the extreme points of C at least when some compactness is assumed and C is recovered from Ext(C) by means of the integral representation theory. In this paper, a more general notion of boundary (Definition 1.I) is considered. Such a boundary needs not contain, or even meet, the extreme points, and thus the classical tools are not available. However, through R. C. James's technique and a remarquable result of S. Simons, a convex set can often be "recovered", in a strong sense, from its boundary (Sect. 1). Tight connections are established between this notion, lacunarity in harmonic analysis (Sect. 2), Banach spaces containing or not ?(N) (Sect. 3), compactness and duality theory (Sect. 4). Finally, Sect. 5 is devoted to miscellaneous results and questions in the spirit of the work, and to examples showing the necessity of the assumptions we made. As a general remark, we did not systematically try to state the results in their most general form, as far as it was not necessary to introduce new ideas for doing so. This is specially true for the results of Sect. 2, where we worked in the duality (2~,~?) of the "little" Fourier analysis although the techniques we used remain valid within the general frame of locally compact abelian groups.