k-Convexity for transferable utility games: several characterizations

The paper deals with the subclass of k-convex n-person transferable utility games which have clear affinities to the well-known convex TU-games. Five new characterizations of a k-convex n-person TU-game are presented in terms of: (1) the unanimity coordinates of TU-game with respect to the particular basis consisting of all n-person unanimity TU-games; (2) the second order partial derivatives of Owen's multilinear extension of the TU-game; (3) the adjusted marginal worth vectors of the TU-game in comparison with the core of the TU-game; (4) a minmodular decomposition of the associated cover TU-game; (5) the concavity of the Lovasz extension of the associated cover TU-game.