4

J.N. BROOKS and J.B. STRANTZEN

Figure 2.

The vector v is chosen to be

parallel to T'nT if T'nT^O

parallel to T' (hence to T) if V n T = 0 .

Thus, in Figure 2, if T and T' are hyperplanes perpendicular to the page, then v will also be

perpendicular to the page. Since v is parallel to T1 and

B2 C /iT' but 6 £ /iT',

it follows that

B2l/ C i/(/iT') but i/(6) £ f(/iT').

Hence i/(6) ^ B2u. It is clear that v(T) is a common hyperplane of support to B\v and Biv at

f(s). To complete the inductive step, we need to show that B\v and B2v satisfy the remaining

conditions of Lemma 1.1.5, namely

if C\ is the frontier of B\v and C2

(*) the frontier of BiVl then C\

is strictly locally inside C2 and C2 is connected.

The need to modify Blaschke's choice

Our choice of is, which is explained later in the proof of Lemma 1.1.5, is guided by the following

considerations involving the concept of a faithful projection.

Definition 1.1.6 The projection v is faithful if the frontier C of Bv consists entirely of points

which are projections of points of S.

If the projection is faithful for both # i and B2 then C\ is strictly locally inside C2. To preserve

the connectedness cf C2 we use a slightly stronger concept, that of finite faithfulness, which will