Ugrás a tartalomhoz

Convex Geometry

Csaba Vincze (2013)

University of Debrecen

2. fejezet - Carathéodory's theorem

2. fejezet - Carathéodory's theorem

In section 1.3 of the previous chapter we defined the affine and the convex hull of a set as the collection of affine and convex combinations of the elements. The structure theorem 1.3.8 of affine sets allows us to generate the affine hull with the help of foundations of classical linear algebra: the problem is how to determine the associated linear subspace of the affinehull. Let H be an arbitrary non-empty subset in the coordinate plane. Theorem 1.3.8 says that the affine hull aff H can be written into the form p+L, where p is in H and L is a uniquely determined linear subspace. Let p in H be an arbitrary point and consider a maximal linearly independent system of vectors

v 1 = - p + w 1 , v 2 = - p + w 2 , , v k = - p + w k ,

(2.1)

where w(1), ..., w(k) are in H. Elements in 2.1 can be interpreted as position vectors of w's with respect to the base point p. Suppose that the orthogonal complement N to the generated linear subspace of 2.1 is spanned by

z k + 1 , , z n .

(2.2)

The element w belongs to the affine hull of H if and only if the position vector w - p is orthogonal to all the vectors z(k+1), ..., z(n). Therefore we have n - k equations

w - p , z k + 1 = 0 , , w - p , z n = 0

(2.3)

to characterize the affine hull of H. The linear independence of the system 2.1 is equivalent to the affinely independence of p, w(1), ... w(k). As another way to prepare (and motivate) the central notion of affinely independence/dependence in the forthcoming sections suppose that

λ 1 v 1 + + λ k v k = λ k + 1 v k + 1 + + λ m v m ,

where both sides of the equation involve combinations of the same type (affine or convex). For the definiteness consider the case of affine combinations:

λ 1 + + λ k = 1   a n d   λ k + 1 + + λ m = 1 .

We have

λ 1 v 1 + + λ k v k + ( - λ k + 1 ) v k + 1 + + ( - λ m ) v m = 0 ,

i.e. the system

v 1 , , v k , v k + 1 , , v m

is linearly dependent in such a way that the sum of the coefficients is equal to zero.

2.1 Affinely dependence and independence

Definition The system

v 1 , , v k

(2.4)

of vectors is affinely dependent if the zero vector can be expressed as a non-trivial linear combination

λ 1 v 1 + + λ k v k = 0

(2.5)

such that the sum of the coefficients is zero:

λ 1 + + λ k = 0 .

(2.6)

The system is affinely independent if it is not affinely dependent.

Remark Affine dependence involves the linear dependence of the system together with an additional requirement 2.6 for the coefficients.

Corollary 2.1.1 The system 2.4 is affinely dependent if and only if

( v 1 , 1 ) , , ( v k , 1 )

is linearly dependent in the coordinate space of dimension n+1.

Proof The existence of a non-trivial scalar k-tuple solving equations 2.5 and 2.6 implies that

λ 1 ( v 1 , 1 ) + + λ k ( v k , 1 ) = ( 0,0 )

(2.7)

and vice versa.

Corollary 2.1.2 Systems containing at least n+2 vectors are affinely dependent.

Proposition 2.1.3 The system 2.4 is affinely independent if and only if for any index i the position vectors

v 1 - v i , , v i - 1 - v i , v i + 1 - v i , , v k - v i

(2.8)

are linearly independent.

Proof Observe that linear combinations of the position vectors 2.8 mean combinations of 2.4 such that the sum of the coefficients is zero and vice versa.

Remark The affine dependence means that we have a non-trivial polygonal chain with sides parallel to the position vectors from one of the elements to the others in the given system (lasso).

Corollary 2.1.4 The system

v 1 , , v k , v k + 1

(2.9)

is affinely independent if and only if the affine hull is of dimension k.

Corollary 2.1.5 Suppose that the system 2.9 is affinely independent. Then any point p of the affine hull has a unique representation as an affine combination of the elements 2.9. The coefficients in this combination are called the affine coordinates of the point p with respect to 2.9.