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 @t(F) of F in terms of n and the independence number @n(F). This question is motivated by a problem of Grunbaum [L. Danzer, B. Grunbaum, 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 @t(F)@?2^n(2nn)(nlogn+loglogn+5n)@n(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. Next, we consider another measure of complexity, the Vapnik-Cervonenkis 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 Grunbaum [B. Grunbaum, 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 Grunbaum, Discrete Comput. Geom. 9 (1) (1993) 101-105] who constructed a counterexample of dual VC-dimension @[email protected]?. Our result implies that no upper bound exists.