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 [L. Danzer, B. Grünbaum, V. Klee, Helly’s theorem and its relatives, in: Proc. Sympos. Pure Math., vol. VII, Amer. Math. Soc., Providence, RI, 1963, pp. 101–180]. Our bound τ ( F ) ≤ 2 n ( 2 n n ) ( n log n + log log n + 5 n ) ν ( F ) is exponential in n , an improvement from the previously known bound of Kim, Nakprasit, Pelsmajer and Skokan [S.-J. Kim, K. Nakprasit, M.J. Pelsmajer, J. Skokan, Transversal numbers of translates of a convex body, Discrete Math. 306 (18) (2006) 2166–2173], which was of order n n . By a lower bound, we show that the right order of magnitude is exponential in n . we consider another measure of complexity, the Vapnik–Červonenkis dimension of F . We prove that  vcdim ( F ) ≤ 3 if n = 2 and is infinite for some F if n ≥ 3 . This settles a conjecture of Grünbaum [B. Grünbaum, Venn diagrams and independent families of sets, Math. Mag. 48 (1975) 12–23]: 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 [D.Q. Naiman, H.P. Wynn, Independent collections of translates of boxes and a conjecture due to Grünbaum, Discrete Comput. Geom. 9 (1) (1993) 101–105] who constructed a counterexample of dual VC-dimension ⌊ 3 n 2 ⌋ . Our result implies that no upper bound exists.