1. M A I N RESUL T A N D S K E T C H P R O O F 1.1 Definitions and Statement of main result To facilitate the statement of our results we introduce some definitions and notation. Throughout, Bt- will be a convex open set in R n , and St- will be the frontier of J3,- for t = 1,2. We suppose st € St and T, is a hyperplane of support to B{ at st-. The open half-space determined by Ti that contains B{ will be denoted by hTi, the other open half space by ~ hT{. Definition 1.1.1 We call the pair of order pairs (si,Ti) and (s2,T2) comparable if T\ and Ti are parallel, and B\ is on the same side of T\ as Bi is of Ti. The unit normals of the hyperplanes of support to £,-, assumed directed towards B,-, will be denoted by VT{. Thus (si,Ti) and (s2,T2) are comparable if and only if I/J^ = VT7- We will also refer to (si,S2) as being a comparable pair of points in this instance. In the following definitions, B$(s2) denotes the closed ball of radius 6 and centre S2. The map t denotes translation by the vector 82 — 8\. Definition 1.1.2 we say Si is locally inside S2 if for each comparable pair of points (si, 52) there is a positive real number 6 such that 57(^Tni(J51)cB2 The definition is illustrated in Figure 1. Figure 1. 2

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