Greedy approximation of characteristic functions

We discuss the problem of sparse representation of domains in $\mathbb R^d$. We demonstrate how the recently developed general theory of greedy approximation in Banach spaces can be used in this problem. The use of greedy approximation has two important advantages: (1) it works for an arbitrary dictionary of sets used for sparse representation and (2) the method of approximation does not depend on smoothness properties of the domains and automatically provides a near optimal rate of approximation for domains with different smoothness properties. We also give some lower estimates of the approximation error and discuss a specific greedy algorithm for approximation of convex domains in $\mathbb R^2$.