yes. and area of the surface of an n-sphere.

http://mathworld.wolfram.com/Hypersphere.html

[math]\sum_{i=1}^{n+1} (x_i-c_i)^2{\leq}r^2[/math] is not the set of all points at a certain distance from the center, but it is a "solid ball", so is it still a n-sphere?

No. You were given the definition of S^n, the n-sphere. commonly we take radius 1 as the standard. With the inequality it is a ball (or sometimes it is called a disc).