Generalized Curvatures Part 5 pps

Chapter 7

Convex Subsets

There is an abundant literature on convexity, crucial in many fields of mathemat￾ics. We shall mention the basic definitions and some fundamental results (without

proof), useful for our topic. In particular, we shall focus on the properties of the

volume of a convex body and its boundary. The reader can consult [9,71,74,79] for

details.

7.1 Convex Subsets

7.1.1 Definition and Basic Properties

Definition 22.

• A subset K of EN is called convex if for all p and q in K, the segment [p,q] lies

in K (Fig. 7.1).

• A compact convex subset K of EN is called a convex body if it has a nonempty

interior (Fig. 7.2).

• The boundary ∂K of a convex body K is called a convex hypersurface.

We shall denote by C (or CEN if we need to be more precise) the space of convex

subsets of EN. We denote by Cb (resp., Cc) the space of convex bodies (resp., com￾pact convex subsets) of EN. The space Cc, endowed with the Hausdorff metric, is a

complete metric space [9, Tome 3,12.9.1.2].

Proposition 1. Let (Kn)n∈N be a sequence of convex subsets of EN whose Hausdorff

limit is the subset K. Then, K is convex.

Moreover, any convex body of EN can be approximated by compact convex poly￾hedra [9, Tome 3,12.9.2.1].

Proposition 2. Let K be a convex body of EN. Then, K is the Hausdorff limit of a

sequence of compact convex polyhedra (Fig. 7.3).

77

78 7 Convex Subsets

. . . . q

p

p

q

Fig. 7.1 The subset on the left is a convex subset of E2

, the one on the right is nonconvex

Fig. 7.2 A convex body K

in E3

K

Fig. 7.3 A sequence of con￾vex subsets Kn in E2 tending

to the (convex) ellipse E

2

K0

K

E

K2

K1

Kn

E

For our purpose, the orthogonal projection (with respect to the usual scalar prod￾uct of EN) is one of the main tools. We mention here classical properties of this map

in the convex context. Basically, we need a deep study of two types of projection:

1. The first one is the projection of a convex subset onto an (affine) subspace of

EN [9, Tome 2,9.11.6 and Tome 3].

Proposition 3. Let P be any vector (or affine) subspace of EN and let K be a

convex subset of EN. Then:

• The orthogonal projection pr(K) of K onto P is convex (Fig. 7.4).

• The map pr : CEN → CP is continuous for the Hausdorff topologies.