Some estimations for the "juxtaposition function" hF and an asymptotic formula for the function hf/hc, where F, G are central symmetric convex bodies, are given. Hadwiger and Grunbaum gave for hp{\) the bounds n2 + n < hp (1) < 3"-1. Grunbaum conjectured (and proved for n = 2 in Pacific J. Math. 11 (1961), 215-219) that for every even r between these bounds there exists in E" an oval F such that hf{\) = r. Lower bounds for hf could be derived in the same way as in Theorems 1 and 2 from a good estimate of packing numbers on a Minkowski sphere, that is, from solutions to a Tammes-type problem in a Banch space.
We give asymptotic estimates for the number of non-overlapping homothetic copies of some centrally symmetric oval B which have a common point with a 2-dimensional domain F having rectifiable boundary, extending previous work of the L.Fejes-Toth, K.Borockzy Jr., D.G.Larman, S.Sezgin, C.Zong and the authors. The asymptotics compute the length of the boundary ∂F in the Minkowski metric determined by B. The core of the proof consists of a method for sliding convex beads along curves with positive reach in the Minkowski plane. We also prove that level sets are rectifiable subsets, extending a theorem of Erdös, Oleksiv and Pesin for the Euclidean space to the Minkowski space.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.