PAR T I LOCAL C O N D I T I O N S FOR C O N T A I N M E N T 0. INTRODUCTION The Rolling Theorems of Blaschke [l] arise from the search for sufficient conditions that the frontier of one convex region in R 2 (or R n ) lies inside the frontier of another convex region. The conditions traditionally used involve comparison of the curvature of the two curves (or hypersurfaces) forming the frontiers of the convex regions, at points where they possess the same unit normals. This, of course, requires that the curves (or hypersurfaces) be twice differentiable. The best results obtained for smooth frontiers in R n are due to J.A. Delgado [5], who works with complete connected oriented hypersurfaces with positive curvatures. Delgado leaves the question open in the case where the 'outside' hypersurface is restricted to have non-negative, rather than positive, curvatures. Wm. J. Firey [6] has obtained general results for convex bodies (that is closures of bounded convex regions) without any smoothness assumptions. In this part of the paper we tackle the problem in full generality, that is, our hypersurfaces consist of frontiers of any convex region in R n . We use geometric rather than analytic techniques and, like Firey, we may not have curvature to work with so we introduce in its place the geometric condition of being 'locally inside'. Application of our result to the case where the hypersurfaces have curvature answers the question posed by Delgado and may also be used to obtain an earlier result of the authors [2]. In substance, it will be shown that 'locally inside' and 'touching' imply 'globally inside'. Received by the editors October 2, 1984 and, in revised form November 3, 1986. 1

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