BLASCHKE'S ROLLING THEOREM IN R1

3

Definition 1.1.3 We say S\ is strictly locally inside S

2

if for each comparable pair of points

(si,s

2

) there is a positive real number 8 such that

ft(*2)nt(£iUSi) C B2U{s2}.

By saying that B\ and Bi have a common hyperplane of support T at s G S\ n 5

2

we will

mean that not only is T a hyperplane of support to both B\ and B

2

at s but also that B\ and £

2

both lie on the same side of T.

We wish to show, in a sense, that 'locally inside' implies 'globally inside'. More specifically we

wish to establish the following theorem, which is our main result.

Theore m 1.1.4 Let Bi be a convex region in R

n

with frontier 5t(t = 1,2). Let s € S\ n Si and

suppose B\ and J32 have a common hyperplane of support T at s. IfSi is locally inside 5

2

and 5

2

is connected and Si n T is bounded, then B\ C J32.

Method of Proof.

In line with the approach taken by Koutroufiotis in [8], our main task will be to establish a

weaker version of the above theorem, namely the following lemma:

Lemm a 1.1.5 Let Bi be a convex region in R n , with frontier 5t(t = 1,2). Let s £ S\ n 5

2

, and

suppose B\ and Bi have a common hyperplane of support T at s. If S\ is strictly locally inside

5

2

and Si is connected, then B\ C B

2

.

With the help of similitudes and other geometric techniques, we shall later use Lemma 1.1.5

to prove Theorem 1.1.4.

Lemma 1.1.5 will be proved by induction using an idea introduced by Blaschke in [1] and

refined by Koutroufiotis in [8]. In essence, if one projects a convex region B in R fc+ 1 orthogonally

onto a k- dimensional affine or hyperplane P with normal v, then the image Bv of B is an open

convex subset of P, and so may be regarded as a convex region in R*. Throughout the paper, when

dealing with projections, we will use v both for the direction of projection and for the projection

map itself, to economise notation.

Our basic aim is: given B\ and Bi in R* + 1 satisfying the conditions of lemma 1.1.5 but not

its conclusions, to produce a projection v such that B\v and Biu, the projections of B\ and Bi in

R*, satisfy the conditions of lemma 1.1.5 but not its conclusions. The inductive hypothesis will

then provide the desired contradiction.

Blaschke fs choice of projection

The principle difficulty in carrying out the inductive step outlined above, lies in the choice of

the direction v of projection. If we were to adopt Blaschke's technique [1] without modification,

we would make the choice in the following way.

Suppose B\ g Bi so that there is a point b G B\ with b g B

2

US

2

. Hence there is a point s' G 5

2

and a hyperplane of support

T1

to 5

2

at s' such that T' separates b from Bi, that is b E ~

hT1.

The relation of b to the hyperplane of support is illustrated in Figure 2 in a cross-sessional view.