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I try to find and prove the VC-dimension of the infinite set of (uni-directional) convex bodies. From intuition, it's clear to me that it goes to infinity, but I don't know the correct way to prove it formally. Any clue for direction?

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For a binary class of data and any number of instances, as you have an infinite number of convex bodies, you can put all instances of a class inside the body and leave others. In this way, you can shatter any set with any arbitrary size. Hence, the VC dimension of an infinite number of convex bodies is infinite.

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