Hausdorff Obstructions to Packing (N-1)-Balls in N-Space
John B. Wilker · Canadian Mathematical Bulletin · 1982
Abstract An arbitrary collection of (N — l)-flats in ℝN is called a frame and an arbitrary assignment of (N — 1))-balls to these (N — 1)-flats is called a loading. Any loading in which the designated (N — 1)-balls are mutually disjoint is called a packing. For the frame consisting of the (N — 1)-flats perpendicular to a given line, every loading is automatically a packing. Although this is obviously not the most general frame to admit a packing, we show two senses in which all frames which admit packings are "at most onedimensional." Our principal tool is the Hausdorff measure-theoretic dimension.