Sequential random packing in the plane

Abstract

The Palasti conjecture on the asymptotic mean proportion of coverage is verified for the sequential random packing of rectangular cars with sides parallel to rectangular boundaries in the models of Rényi and Solomon. The extension to n dimensions is given. An extension to a random car size model is indicated.

Reply to Letters of M. Tanemura and E. M. Tory and D. K. Pickard

“…If the (2a ) x (2{:3) rectangles do not cut the line I, every space which can accommodate a car must eventually be filled, and the horizontal placement is independent of the vertical placement. The arguments of the two letters do not appear to refute the results of [2].…”

“…These changes do not alter the results of [2]. The phrase ([2], p. 806, above (2.8a», 'From the independence of x, y coordinates...', refers to attempted placements, as Tory and Pickard indicate.…”

On Weiner's Proof of the Palásti Conjecture

“…In a recent paper Weiner (1978) claims to have proved the Palasti conjecture (see Palasti (1960)) respecting the asymptotic mean density of random sequential packing in the plane or the higher-dimensional space. This conjecture has previously been tested by the use of Monte Carlo simulation techniques.…”

Has the Palásti Conjecture Been Proved?: A Criticism of a Paper with H. J. Weiner

“…In opposition to this, I have great doubt about his conclusions. The purpose of this note is to point out the most fundamental errors in the paper of Weiner (1978). In this note, I mention only Model I (Renyi's car parking problem) because Weiner's conclusions on Model II and on random size cars are based essentially on the same errors as those contained in the conclusions of Model I.…”

“…Has the Paldsti conjecture been proved? : a criticism of a paper by H. J. Weiner Weiner (1978) claims that he has verified the Palasti conjecture (Palasti (1960)) in the plane and, more generally, in the n-dimensional space. In opposition to this, I have great doubt about his conclusions.…”