Remark 3.4.4 Perhaps the most important reason for these precautions is the danger of so-called overflows. When a computer is asked to do an arithmetic operation whose
III. Convex Sets
1.1 Definition and First Examples
Definition 1.1.1 The set C
c
lRn is said to be convex if ax+
(l - a)x' is in C whenever x and x' are in C, and a E ]0, I [ (or equivalently a E [0, ID.
0Geometrically, this means that the line-segment
[x, x'] := {ax
+
(1 - a)x' :°
~ a ~ I}is entirely contained in C whenever its endpoints x and x' are in
c.
Said otherwise: the set C - {e} is a star-shaped set whenever c E C (a star-shaped set is a set containing the segment [0, x] for all its points x). A consequence of the definition is that C is also path-connected, i.e. two arbitrary points in C can be linked by a continuous path.Examples 1.1.2 (Sets Based on Affinity) We have seen in Chap. I that the convex sets in lR are exactly the intervals; let us give some more fundamental examples in several dimensions.
(a) An affine hyperplane, or hyperplane, for short, is a set associated with (s, r) E
IRn x IR (s
¥:
0) and defined byHs.r := {x E JRn : (s, x) = r}.
An affine hyperplane is clearly a convex set. Fix s and let r describe JR ; then the affine hyperplanes Hs.r are translations of the same linear, or vector, hyperplane Hs.o. This Hs.o is the subspace of vectors that are orthogonal to s and can be denoted by Hs.o = {s}.!.. Conversely, we say thats is the normal to Hs.o (up to a multiplicative constant).
Affine hyperplanes play a fundamental role in convex analysis; the correspondence between 0
i=
s E JRn and HS •I is the basis for duality in a Euclidean space.(b) More generally, an affine subspace, or affine manifold, is a set V such that the (affine) line {ax
+
(1 - a) x' : a E JR) is entirely contained in V whenever x and x' are in V (note that a single point is an affine manifold). Again, an affine manifold is clearly convex.Take v E V; it is easy - but instructive - to show that V - {v} is a subspace ofJRn , which is independent of the particular v; denote it by Yo. Thus, an affine manifold V is nothing but the translation of some vector space Yo, sometimes called the direction ( -subspace) of V. One can therefore speak of the dimension of an affin manifold V: it is just the dimension of Yo. We summarize in Table 1.1.1 the particular cases of affine manifolds.
Table 1.1.1. Various affine manifolds
Name Possible definition Direction Dimension
point {x} (x E ]Rn) to} 0
affine {axl
+
(1-a)x2 : a E 1R} vector line line XI -:f: X2 (both in IRn) lR(x - x') affine {x E IRn : (s.x) = r} vector byperpl.hyperplane (s -:f: O. r E 1R) {s }.l n-l
(c) The half-spaces ofll~n are those sets attached to (s. r) E JRn x JR, s
i=
0, and defined by{x E JRn : (Sf x) ~ r} (closed half-space) {x E JRn : (Sf x) < r} (open half-space);
"affine half-space" would be a more accurate terminology. Naturally, an open [resp.
closed] half-space is really an open [resp. closed] set; it is the interior [resp. closure]
of the corresponding closed [resp. open] half-space; and the affine hyperplanes are the boundaries of the half-spaces; all this essentially comes from the continuity of the
scalar product (s • . ). 0
Example 1.1.3 (Simplices) Call a = (al •...• ak) the generic point of the space IRk. The unit simplex in IRk is
Llk:={aEIRk : :Er=laj=l. a j ;;::Ofori=l •...• k}.
1 Generalities 89 Equipping ]Rk with the standard dot-product, {el' ... ,ek} being the canonical basis and e :
=
(1, ... , 1) the vector whose coordinates are all 1, we can also writeLlk:={aE]Rk: eTa=l,
eJa~Ofori=l,
...,k}.
(1.1.1)Observe the hyperplane and half-spaces appearing in this definition. Unit simplices are convex, compact, and have empty interior - being included in an affine hyperplane.
We will often refer to a point a E Llk as a set of (k) convex multipliers.
It is sometimes useful to embed Llk in]Rm, m > k, by appending m - k zeros to the coordinates of a E ]Rk, thus obtaining a vector of Llm . We mention that a so-called simplex of]Rn is the figure formed by n
+
I vectors in "nondegenerate positions"; in this sense, the unit simplex of]Rk is a simplex in the affine hyperplane of equation eTa=
1; see Fig. 1.1.1.a:J
~. -~'2
~63
Fig.I.I.I. Representing a simplex
If we replace eTa = 1 in (1.1.1) by eTa:::; 1, we obtain another important set, convex, compact, with nonempty interior:
Ll~
:= {a E]Rk : eTa~
1, aj~
0 for i = 1, ... ,k} .
In fact, this set can also be described as follows:
a E Ll~ {=> 3ak+1 ~ 0 such that (a, ak+l) E Llk+l .
In this sense, the simplex Ll~ C ]Rk can be identified with Llk+l via a projection operator.
A (unit) simplex is traditionally visualized by a triangle, which can represent ,,13
or Lli; see Fig. 1.1.1 again. 0
Example 1.1.4 (Convex Cones) A cone K is a set such that the "open" half-line {ax: a > O} is entirely contained in K whenever x E K. In the usual representation of geometrical objects, a cone has an apex; this apex is here at 0 (when it exists: a subspace is a cone but has no apex in this intuitive sense). Also, K is not supposed to contain 0 - this is mainly for notational reasons, to avoid writing 0 x (+00) in some situations. A convex cone is of course a cone which is convex; an example is the set defined in ]Rn by
(Sj, x) = 0 for j = 1, ... , m, (sm+j'x) ~ 0 for j = 1, ... , p, (1.1.2) where the S j 's are given in jRn (once again, observe the hyperplanes and the half-spaces appearing in the above example, observe also that the defining relations must have zero right-hand sides).
Convexity of a given set is easier to check if this set is already known to be a cone: in view of Definition 1.1.1, a cone K is convex if and only if
x+YEK whenever xandyareinK,
i.e. K
+
K C K. Subspaces are particular convex cones. We leave it as an exercise to show that, to become a subspace, what is missing from a convex cone is just symmetry (K = - K).A very simple cone is the nonnegative orthant of jRn
Q+:={x=(~t, ... ,~n): ~j~Ofori=I, ... ,n}.
It can also be represented in terms of the canonical basis:
Q+={I:l=tajej: aj~Ofori=l, ...
,n}
or, in the spirit of (1.1.2):
Q+={XEjRn: (ej,x}~Ofori=l, ... ,n}.
Convex cones will be of fundamental use in the sequel, as they are among the simplest convex sets. Actually, they are important in convex analysis (the "unilateral"
realm of inequalities ),just as subspaces are important in linear analysis (the "bilateral"
realm of equalities). 0