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

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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?

1Hi and welcome to this community! I've noticed that you have asked several questions on this site and I really appreciate that, but you should not ask for solutions without even explaining what you have tried so far. You should also try to ask one question per post. I suggest you ask the first question in this post and create another post for the second question ;) – nbro – 2020-01-16T13:31:52.263

@nbro Thank you for your comment. I will edit my post. – j200932 – 2020-01-16T14:13:12.370