The Collaborative International Dictionary of English v.0.48:

hypersphere \hy"per*sphere\ (h[imac]"p[~e]r*sf[=e]r), n.
   A mathematical object existing in more than three dimensions,
   analogous to the sphere in that all points on the surface are
   equidistant from the central point; a generalization of a
   sphere in more than three dimensions.
   [PJC]