Abstract: Non-convex sets in linear spaces are divided to sets which are finite unions of convex sets, sets which are countable unions of convex set and all others.
A hierarchy of length omega1 of non-convexity of closed sets in separable Banach spaces will be presented, which refines the class of sets which are countable unions of convex sets.
First the problem of covering by convex sets is reduced to computing the chromatic number of a certain hypergraph. Using topology, an ordinal rank of a set is defined and it is seen that a bounded rank is sufficient for countable chromatic number. Finally, in Polish linear spaces, the sufficient condition is also necessary. Thus, to every closed set in a Polish linear space there corresponds an ordinal rho so that the set is a countable union of convex sets if and only if rho is countable. This ordinal is the "non-convexity degree" of the set.
A similar hierarchy was previously known for closed sets in the plane.