  In this paper we prove that if $G$ is a p-torus (resp. torus) group acting
without fixed points on a finitistic space X (resp. with finitely many orbit
types), then the G-index $i_G(X) < 1$. Using this G-index we obtain a
generalization of the Central Point Theorem and also of the Tverberg Theorem
for any d-dimensional Hausdorff space.
