Parallel Packing Squares into a Triangle

Assume that $$T_h$$ T h is a triangle with the interior angles at the base of the measure not greater than $$90^0$$ 90 0 , with the base length 1 and the height h. Let S be a square with a side parallel to the base of $$T_h$$ T h and let $$\{S_n\}$$ { S n } be a collection of the homothetic copies of S. A tight upper bound of the sum of the areas of squares from $$\{S_n\}$$ { S n } that can be parallel packed into a triangle $$T_h$$ T h is determined.

Van der Waerden's Theorem on Homothetic Copies of {1,1+s,1+s+t}

For all positive integers

Asymptotics of generalized Hadwiger numbers

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 L. Fejes Tóth, K. Böröckzy 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 Erdős, Oleksiv and Pesin for the Euclidean space to the Minkowski space.

On the Bezdek–Pach Conjecture for Centrally Symmetric Convex Bodies

The Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ℝd is 2d. Naszódi proved that the quantity in question is not larger than 2d+1. We present an improvement to this result by proving the upper bound 3 · 2d–1 for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.