Hypersphere - Pattern matching using Centroid, Radius and Diameter

Question

I have a hyper-sphere formed with set of $n$-dimensional data points. I could calculate centroid ($X_0$), radius($R$) and diameter($D$).

Using these $X_0, R, D$, how I can find whether the a given data point falls within the boundary or outside.

Would like to know if any other measure caters this use-case.

Answer 1

The implicit equation of an hypersphere is

$$(x-x_o)^2+(y-y_o)^2+\dots(t-t_o)^2=r^2.$$

Inside the hypersphere, $LHS<RHS$ and conversely outside.