Our main observation concerns closed geodesics on surfaces M with a smooth Finsler metric, i.e. a function F : TM → [0,∞) which is a norm on each tangent space TpM , p ∈ M , which is smooth outside of the zero section in TM , and which is strictly convex in the sense that Hess(F ) is positive definite on TpM \ {0}. One calls a Finsler metric F symmetric if F (p,−v) = F (p, v) for all v ∈ TpM . ...

In [BL] in relation to the unsolved Bang’s plank problem (1951) we obtained a lower bound for the sum of relevant measures of cylinders covering a given d-dimensional convex body. In this paper we provide the packing counterpart of these estimates. We also extend bounds to the case of r-fold covering and packing and show a packing analog of Falconer’s results ([Fa]).

In 1950, C.A. Rogers introduced and studied the simultaneous packing and covering constants for a convex body and obtained the first general upper bound. Afterwards, they have attracted the interests of many authors such as L. Fejes Tóth, S.S. Rys̆kov, G.L. Butler, K. Böröczky, H. Horváth, J. Linhart and M. Henk since, besides their own geometric significance, they are closely related to the pac...

Abstract One of the basic assumptions in hub covering problems is considering the covering radius as an exogenous parameter which cannot be controlled by the decision maker. Practically and in many real world cases with a negligible increase in costs, to increase the covering radii, it is possible to save the costs of establishing additional hub nodes. Change in problem parameters during the pl...