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 Rn. 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 [2]. Our bound τ(F) ≤ 2n (2n n ) (n logn+ log logn+ 5n)ν(F) is exponential in n, an improvement from the previously known bound of Kim, Nakprasit, Pelsmajer and Skokan [10], which was of order nn. 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–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 Günbaum [6]: Show that the maximum dual VC-dimension of a family of positive homothets of a given convex body K in Rn is n + 1. This conjecture was disproved by Naiman and Wynn [13] who constructed a counterexample of dual VC-dimension ⌊ 3n 2 ⌋ . Our result implies that no upper bound exists. 1. Definitions and Results A convex body in R is a compact convex set with non-empty interior. A positive homothet of a set S ⊆ R is a set of the form λS + x, where λ > 0 and x ∈ R. The cardinality, closure, convex hull and volume of S are denoted as card(S), cl(S), conv(S) and vol(S), respectively. The origin of R is denoted o. Let F be a family of positive homothets (or translates) of a given convex body K in R. In this note we study two approaches to measuring the complexity of F . First, we bound the transversal number τ(F) in terms of the dimension n and the independence number ν(F). The transversal number τ(F) of a family of sets F is defined as τ(F) = min {card(S) : S ∩ F 6= ∅ for all F ∈ F}. The independence number ν(F) of F is defined as ν(F) = max {card(S) : S ⊆ F and S is pairwise disjoint}. Clearly ν(F) ≤ τ(F). The problem of finding an inequality in the reverse direction originates in the following question of Grünbaum [2]: Is it true that ν(F) = 1 Date: 2009 July 29. 2000 Mathematics Subject Classification. Primary 52A35, Secondary 05D15.