Abstract: | For let be the -ideal in generated by all sets which do not contain equidistant points in the usual metric on . For each a set is constructed in so that the -ideal which is generated by the convex subsets of restricted to the convexity radical is isomorphic to . Thus is equal to the least number of convex subsets required to cover -- the convexity number of . For every non-increasing function we construct a model of set theory in which for each . When is strictly decreasing up to , uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of . It is conjectured that , but never more than , different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of . This conjecture is true for and . |