2.1 The Relative Interior
Let C be a nonempty convex set in Rn. If int C =1= !2J, one easily checks that the affine hull aff C is the whole of Rn (because so is the affine hull of a ball contained in C):
we are dealing with a "full dimensional" set. On the other hand, let C be the sheet of paper on which this text is written. Its interior is empty in the surrounding space R3,
but not in the space R2 of the table on which it is lying; by contrast, note that cl Cis the same in both spaces.
This kind of ambiguity is one of the reasons for introducing the concept of relative topology: we recall that a subset A ofRn can be equipped with the topology relative to A, by defining its "closed balls" R(x, 8)nA, for x E A; then A becomes a topological space in its own. In convex analysis, the topology of Rn is of moderate interest: the topologies relative to affine manifolds turn out to be much richer.
2 Convex Sets Attached to a Convex Set 103 Definition 2.1.1 The relative interior ri C (or relint C) of a convex set C C ]Rn is the interior of C for the topology relative to the affine hull of C. In other words: x E ri C if and only if
X E aff C and 38 > 0 such that (aff C)
n
B(x, 8)c
C.The dimension of a convex set C is the dimension of its affine hull, that is to say
the dimension of the subspace parallel to aff C. 0
Thus, the wording "relative" implicitly means by convention "relative to the affine hull".
Of course, note that ri C C C. All along this section, and also later in Theorem Y.2.2.3, we will see that aff C is the relevant working topological space. Already now, observe that our sheet of paper above can be moved ad libitum in]R3 (but not folded: it would become nonconvex); its affine hull and relative interior move with it, but are otherwise unaltered. Indeed, the relative topological properties of C are the properties of convex sets in ]Rk, where k is the dimension of C or aff C. Table 2.1.1 gives some examples.
Table 2.1.1. Various relative interiors
C affC dimC riC
{x} {x} 0 {x}
[x, x'] affine line ]x, X/[
x =1= x' generated by x and x'
L\n of equation eTa affine manifold
=
I n-l {a E L\n : ai > O}B(XQ,8) ]Rn n int B(xQ, 8)
Remark 2.1.2 The cluster points of a set C are in aff C (which is closed and contains C), so the relative closure of C is just cl C: a notation relcl C would be superfluous. On the contrary, the boundary is affected, and we will speak of relative boundary:
rbd C := cl C\ ri C . o
A first demonstration of the relevance of our new definition is the following:
Theorem 2.1.3
If
C#
0, then ri C#
0. In jact, dim(ri C)=
dim C.PROOF. Let k := 1
+
dim C. Since aff C has dimension k - 1, C contains k elements affinely independent Xl. •.• , Xk. Call L\ := CO{XI' .•. ,x,d
the simplex that they gen-erate; see l'lg. 2.1.1; aff L\=
at! (; because Ll C C and dim L\=
k - 1. The proof will be finished if we show that L\ has nonempty relative interior.Take
x
:= Ilk 2:7=1Xi
(the "center" of L\ ) and describe aff L\ by points of the formk k
x+y=x+ Lai(Y)Xi = L[t+ai(Y)]Xi,
i=1 i=1
affC
Fig.2.1.1. A relative interior is nonempty
where a(y) = (al(y), ... , ak(Y)) E IRk solves
k
LaiXi = y, i=l
k
Lai=O.
i=l
}-Because this system has a unique solution, the mapping Y ~ a(y) is (linear and) continuous: we can find 8 > 0 such that
lIyll ::;;
8 implieslaj(y)l::;; 11k for i
=
1, ... , k, hence i+
y E .d.In other words, i E ri.d erie.
It follows in particular dim ri C = dim.d = dim
c. o
Remark 2.1.4 We could have gone a little further in our proof, to realize that the relative interior of L1 was
{L~=l aixi : L~=l ai
=
1, ai > 0 for i = 1, ... ,k}.
Indeed, any point in the above set could have played the role of i in the proof Note, incidentally, that the above set is still the relative interior of CO{XI, ... , Xk}, even ifthe Xj'S are not affinely
independent. 0
Remark 2.1.5 The attention of the reader is drawn to a detail in the proof of The-orem 2.1.3: .d C C implied ri..1 erie because .d and C had the same affine hull, hence the same relative topology. Taking the relative interior is not a monotone opera-tion, though: in JR, {OJ C [0, 1] but {OJ = ri{O} is not contained in the relative interior
]0, l[ of [0, 1]. 0
We now tum to a very usefol technical result; it refines the intermediate result in the proof of Proposition 1.2.7, illustrated by Fig. 1.2.3: when moving from a point in ri C straight to a point of cl C, we stay inside ri
c.
2 Convex Sets Att&ched to a Convex Set 105 Lemma 2.1.6 Let x E cl C and x' EriC. Then the half-open segment
lx, x']
=
{ax+
(1 - a)x' :°
~ a < l}is contained in ri C.
PROOF. Take x"
=
ax+
(1-a)x', with 1 > a ~ 0. To avoid writing"n
aft' C" every time, we assume without loss of generality that aft' C = ]Rn.Since x E cl C, for all 8 > 0, x E C
+
B(O, 8) and we can write B(X",8)=
ax + (1 - a)x' + B(O, 8)C aC + (1 - a)x' + (1
+
a)B(O, 8)=
aC+
(1 - a){x'+
B(O, :~~8)}.Since x' E intC, we can choose 8 so small that x'
+
B(O, ~8} C C. Then we have B(X",8) C aC+
(1 - a)C = C(where the last equality is just the definition of a convex set).
o
Remark 2.1.7 We mention an interesting consequence of this result: a half-line issued from x' E ri C cannot cut the boundary of C in more than one point; hence, a line meeting ri C cannot cut cl C in more than two points: the relative boundary of a convex set is thus a fairly regular object, looking like an "onion skin" (see Fig. 2.1.2). 0
/x.~ ~~X
Fig. 2.1.2. The relative boundary of a convex set
Note inparticularthat[x, x'] eriC whenever x and x' areinri C, which confirms that ri C is convex (cf. Proposition 1.2.7). Actually, ri C, C and cl C are three convex sets very close together: they are not distinguished by the operations "aft''', "ri" and
"cl" .
Proposition 2.1.8 The three convex sets ri C, C and cl C have the same affine hull (and hence the same dimenSion), the same relative interior and the same closure (and hence the same relative boundary).
PROOF. The case of the affine hull was already seen in Theorem 2.1.3. For the others, the key result is Lemma 2.1.6 (as well as for most other properties involving closures and relative interiors). We illustrate it by restricting our proof to one of the properties, say: ri C and C have the same closure.
Thus, we have to prove that cl C C cl(ri C). Let x E cl C and take x' EriC (it is possible by virtue of Theorem 2.1.3). Because ]x, x'] eriC (Lemma 2.1.6), we do have that x is a limit of points in ri C (and even a "radial" limit); hence x is in the
closure of ri C. 0
Remark 2.1.9 This result gives one more argument in favour of our relative topology: if we