In game theory, a convex game is one in which the incentives for joining a coalition increase as the coalition grows. This paper shows that the core of such a game--the set of outcomes that cannot be improved on by any coalition of players--is quite large and has an especially regular structure. Certain other cooperative solution concepts are also shown to be related to the core in simple ways: (1) The value of a convex game is the center of gravity of the extreme points of the core. (2) The von Neumann-Morgenstern stable set solution of a convex game is unique and coincides with the core. Similar results for (3) the kernel and (4) the bargaining set will be presented in a later paper. Here, it is also shown that convex games are not necessarily the sum of any number of convex measure games. 37 pp. Ref.