  We study a game where two players take turns selecting points of a convex
geometry until the convex closure of the jointly selected points contains all
the points of a given winning set. The winner of the game is the last player
able to move. We develop a structure theory for these games and use it to
determine the nim number for several classes of convex geometries, including
one-dimensional affine geometries, vertex geometries of trees, and games with a
winning set consisting of extreme points.
