An on-line potato-sack theorem

Abstract

We discuss packings of sequences of convex bodies of Euclidean n-space En in a box and particularly in a cube. Following an Auerbach-Banach-Mazur-Ulam problem from the well-known Scottish Book, results of this kind are called potato-sack theorems. We consider on-line packing methods which work under the restriction that during the packing process we are given each succeeding "potato" only when the preceding one has been packed. One of our on-line methods enables us to pack into the cube of side d>1 in En every sequence of convex bodies of diameters at most 1 whose total volume does not exceed ( {Mathematical expression}). Asymptotically, as d→∞, this volume is as good as that given by the non-on-line methods previously known. © 1991 Springer-Verlag New York Inc.