Tight bounds on discrete quantitative Helly numbers

Quantitative Tverberg Theorems Over Lattices and Other Discrete Sets

“…We end by remarking that using Theorem 3 of the recent paper [6] one can derive a similar result as our Theorem 1.6.…”

“…Corollary 1.1 follows from Theorem 1.4, by setting S = Z d and applying the results in result of Averkov et al [6], as will be discussed further in Section 2. The special case of S = Z d and k = 1 was considered previously in Eckhoff [22] and in the case of m = 2 (Radon partitions) by Onn [28].…”

“…, 4. Most recently, Averkov et al [6] have expressed the quantitative S-Helly number H S (k) in terms of polytopes with vertices in S containing exactly k points of S in their interior.…”