N-sphere math problem

No, the n-sphere is the surface defined by the equality.

 

Just discovered that geometers and topologists use different convention for the naming :
http://mathworld.wolfram.com/Hypersphere.html

 

All points on [itex]\sum_{i=1}^{n+1} (x_i-c_i)^2 = r^2[/itex] belong in an n-sphere (not points inside).

 

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.

 

How about "interior of the (n+1)-ball"?

 

The convention in topology as I learned it is

[tex]\sum_{i=1}^{n} (x_i-c_i)^2{\leq}r^2[/tex]
is a closed n-ball or radius r (or just "closed ball" when you're in R^n Euclidean space)

[tex]\sum_{i=1}^{n} (x_i-c_i)^2< r^2[/tex]
is an open n-ball or radius r or "open ball" in R^n.

Note that in R^n, a ball is n-dimensional and a sphere is (n-1)-dimensional; i.e., a topologist's n-sphere is the boundary of an (n+1)-ball.

 

The interiot is commonly called a ball or a disc.