Defining an n-dimensional orthogonal basis set I have a set of n-dimensional points. I am performing clustering analysis on them to partition them into groups. Now, I would like to find a basis set to be able to describe the layout of the groups' borders.
For example, if my points were only three dimensional, then I could easily use spherical harmonics to describe the groups. If there are, say, five groups then each group can be written as a linear combination of spherical harmonics, and I can easily define the borders of each group.
I cannot, however, use spherical harmonics on, for example, a five dimensional set of points, can I? If not, is there a family of basis sets that can define shapes in n-dimensions?
 A: I am sorry to answer with a question here, but it seems I cannot comment your question as others did.
I am not fully understanding what you mean by "borders" of the groups. What exactly do you mean?
As I understand it, if each cluster is a cloud of points, then you could get the principal axis of that cloud (SVD, PCA, moment of inertia, you name it) and build the smallest ellipsoid (this may not enclose the points but it will be a more or less nice description of their size).
If you really want something enclosing the clouds then you can use the convex-hull as somebody suggested above or if you want non-convex borders you can get the alpha-shape of the clouds. I do not know many implementations of alpha-shapes: CGAL or the equivalent in MATLAB (it is using CGAL code, but I do not know since which version). I wouldn't know any implementation of alpha-shapes for more than 3 dimensions, not even know whether that is possible, but I guess it should.  A link is in order, here is the 2D alpha-shape in CGAL
A: Well, spheres exist in arbitrary dimensions.
In 1d, they are intervals. In 2D they're circles. In 3D they're the obvious sphere. But there exist hyperspheres in arbitrary dimensions.
They also exist for other distance functions. Consider a sphere to be defined as the set of points $x$ with $d(o,x) \{<,=,\leq\} r$ depending on which kind of sphere you want (hollow, etc).
Quoting Wikipedia:

The classical spherical harmonics are defined as functions on the unit sphere $S_2$ inside three-dimensional Euclidean space. Spherical harmonics can be generalized to higher dimensional Euclidean space $R^n$ as follows. [...]

