[tex]\sum_{i=1}^{n+1} (x_i-c_i)^2{\leq}r^2[/tex] is not the set of all points at a certain distance from the center, but it is a "solid ball", so is it still an n-sphere?
I guess you'd simply call them "points inside the n-sphere" (?), though I'm not sure if 'inside' is well defined. In this case, though, you could define 'inside' as being on the same side of the surface as the center.