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On the Transversal Number and VC-Dimension of Families of Positive Homothets of a Convex Body

Let F be a family of positive homothets (or translates) of a given convex body K in R^n. We investigate two approaches to measuring the complexity of F. First, we find an upper bound on the transversal number $τ(F)$ of F in terms of $n$ and the independence number $ν(F)$. This question is motivated by a problem of Grünbaum. Our bound $τ(F)\le 2^n \binom{2n}{n} (n \log n + \log\log n + 5n) ν(F)$ is exponential in n, an improvement from the previously known bound of Kim, Nakprasit, Pelsmajer and Skokan, which was of order n^n. By a lower bound, we show that the right order of magnitude is exponential in n. Next, we consider another measure of complexity, the Vapnik--Chervonenkis dimension of F. We prove that this quantity is at most 3 if n=2 and is infinite for some F if n>2. This settles a conjecture of Günbaum: Show that the maximum dual VC-dimension of a family of positive homothets of a given convex body K in R^n is n+1. This conjecture was disproved by Naiman and Wynn, who constructed a counterexample of dual VC-dimension $\lfloor 3n/2\rfloor$. Our result implies that no upper bound exists.