# How can I show that the VC dimension of the set of all closed balls in $\mathbb{R}^n$ is at most $n+3$?

How can I show that the VC dimension of the set of all closed balls in $$\mathbb{R}^n$$ is at most $$n+3$$?

For this problem, I only try the case $$n=2$$ for 1. When $$n=2$$, consider 4 points $$A,B,C,D$$ and if one point is inside the triangle formed by the other three, then we cannot find a circle that only excludes this point. If $$ABCD$$ is convex assume WLOG that $$\angle ABC + \angle ADC \geq 180$$ then use some geometric argument to show that a circle cannot include $$A,C$$ and exclude $$B,D$$.

For the general case I’m thinking of finding $$n+1$$ points so that a ball should be quite ‘large‘ to include them, and that this ball can not exclude the other 2 points. However, in high-dimensional case I do not know how to use maths language to describe what is ‘large’.

Can anyone give some ideas to this question please?