be explained later. The need for faithful projections is not surprising since comparable pairs in
relation to C\ and C2 will need to be "lifted" to comparable pairs in relation to S\ and 52.
At the degree of generality which we seek, examples exist which indicate that the classical
choice of v need not guarantee a faithful projection, at least in the reduction from R
to R
. To
make allowance for this possibility, we are forced to make a slightly different choice of direction i/,
which complicates the argument to some extent.
The case k - 2
The induction reduces Lemma 1.1.5 to the case k = 2. This case is handled by representing
'substantial' portions of S\ and 5*2, or C\ and C2 as we refer to them in R
, by graphs of functions
/ i : Ji - R and f2 : J2 -* R.-
Here J\ and J2 are intervals containing 0 while / i and f2 are convex non-negative functions
satisfying /x(0) = /
(0) = 0.
The condition that C\ is strictly locally inside C2 gives rise to a simple condition on / i and
/2, which in turn implies fi(x) f2{x) for each non-zero x in J\ n J2. This latter result is then
exploited by means of translations, to obtain a number of results about f\ and f2. The results
essentially amount to the fact that the support function of B\, is never greater than the support
function of B2i even though we do not specifically write in these terms but rather directly show
Bi C B2.
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