A point of a convex set S is said to be extreme (for S) if it is not an interior point of any line segment contained in S. For instance, if S is a closed convex polygon in the plane, then the extreme points of S are precisely its vertices. Also, if S is a closed ball in the usual Euclidean space – think of an orange or a watermelon – then every point of the boundary sphere is extreme for S (and these are the only extreme points).
However, the situation may change dramatically if we consider balls in more general normed vector spaces. In particular, various function spaces are worth looking at. One striking example is offered by L1, the space of integrable functions on the interval [0,1], say. The (closed) unit ball of L1 is the set of those functions f for which the integral of |f| over [0,1] does not exceed 1, and it turns out that this set has no extreme points at all. Thus, the unit ball of L1 is no longer “round”; it is actually pretty “flat”!
Now, we turn to more sophisticated function spaces whose members have prescribed gaps (or holes) in their Fourier spectra. This means that only certain selected harmonics show up in the function’s Fourier series, while others are forbidden. Our lacunary function spaces are then normed appropriately (the norm being either uniform or inherited from L1), and we determine the extreme points of the unit balls that arise.