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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