Abstract
Blaschke's rolling theorem in its orginal form dealt with the problem of finding sufficient
conditions for a ball D\ to roll freely inside a compact convex set Di with non-empty interior.
Here, D\ is said to roll freely inside Di if at each frontier point 52 of Di there is a translation t
such that t(Di) and Di share a common tangent hyperplane at si and t(D\) is contained in D%.
Wm. J. Firey introduced a natural extension of the concept of free rolling whereby D\ is also a
compact convex set. The main thrust of this paper is to examine the problem in full generality in
R
n
.
The paper is divided into two parts. Part I is primarily concerned with producing conditions of
a local nature sufficient to ensure an arbitrary open convex set B\ is a subset of another arbitrary
open convex set Bi. These results are self contained, and are of interest in themselves, as they
follow other contributions on the same theme in the literature from people such as Koutroufiotis,
Rauch and Delgado. Part I also sets up the machinery for Part II, where necessary and sufficient
conditions of a local nature are found for the general free rolling problem. Part II also contains
information regarding the nature of the intersection of the frontiers of B\ and Bi given that they
satisfy the conditions for containment derived in Part I, these results being in line with the works
of Koutroufiotis and Delgado.
More detail of content appears in the introductions to the individual parts.
Part I has been produced under the joint authorship of J.N. Brooks and J.B. Strantzen and
Part II under the sole authorship of J.B. Strantzen.
Ke y Words and Phrase s free rolling, upper indicatrix, upper radius of curvature, touching
convex regions, outer osculating radius, upper principal radius of curvature.
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