N-sphere math problem

Main Question or Discussion Point

[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?
 

Answers and Replies

Gokul43201
Staff Emeritus
Science Advisor
Gold Member
6,987
14
No, the n-sphere is the surface defined by the equality.
 
who what is the inequality?
 
Gokul43201
Staff Emeritus
Science Advisor
Gold Member
6,987
14
Gokul43201
Staff Emeritus
Science Advisor
Gold Member
6,987
14
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 know that, i was asking what the points inside(the inequality) is called, since it isn't a sphere.
 
Gokul43201
Staff Emeritus
Science Advisor
Gold Member
6,987
14
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.
 
Last edited:
the volume which said n-sphere encapsulates?
 
robphy
Science Advisor
Homework Helper
Insights Author
Gold Member
5,399
676
How about "interior of the (n+1)-ball"?
 
rachmaninoff
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.
 
matt grime
Science Advisor
Homework Helper
9,394
3
The interiot is commonly called a ball or a disc.
 

Related Threads for: N-sphere math problem

  • Last Post
Replies
8
Views
747
  • Last Post
Replies
9
Views
5K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
990
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
2
Views
3K
Top