• affS:=T{M ⊆E:S ⊆M and M is affine}, called the affine hullof S;
• convS:=T
{C ⊆E:S⊆C andC is convex}, called theconvex hull ofS.
Even though the above operations will not be used very often in the remainder of the text, they are important in some results and definitions that we see in this chapter. However, one may find it hard to have much intuition on the hull operations with the definitions given above. The following result shows an easier way to see the hull operations: the affine (convex) hull of a set S⊆E is the set of all affine (convex) combinations of finite subsets of points inS.
Proposition 3.1.2 (see [59, Chapter 1] and [59, Theorem 2.3]). For any S⊆E, we have affS =
nXm
i=1
λixi ∈E:for every m∈N, every{xi}mi=1 ⊆S, and everyλ∈Rm s.t.
m
X
i=1
= 1 o
and convS=
nXm
i=1
λixi∈E:for every m∈N, every{xi}mi=1⊆S, and every λ∈Rm+ s.t.
m
X
i=1
= 1 o
. For concreteness, let us look at a small example. Define S := {(1,0),(2,1)} ⊆ R2. With the above proposition, one can easily see thatconvS is the line segment between the points (1,0)and (2,1)and thataffS is the line that passes through (1,0) and(2,1). By setting S0 :=S∪ {(2,0)}, we have thatconvS0 is the region enclosed by the triangle formed by the points inS0 and, interestingly, affS0 is the entire space R2.
the 2-dimensional case. This is where the notion of relative interior comes in handy, since it is built exactly for the purpose of looking at the interior the set would have if it were “full-dimensional”, that is, in the case it lived in a space in which it had non-empty interior. In the example of the line segment, we haveriC = 0⊕(0,1) ={(0, λ) : λ∈(0,1)}. On Figure 3.2 we illustrate a 3-dimensional example.
Figure 3.2: Illustration of the relative interior (yellow) and affine hull (red) of the setC, a closed two-dimensional rectangle in R3, whose interior is empty.
It is worth warning that the operation of taking relative interior of sets can sometimes behave in unexpected ways. For example, if C ⊆ E and D ⊆ E are such that C ⊆ D, then from the definitions of closure and interior one can see that clC ⊆clD and that intC ⊆intD. However, riC is not necessarily contained in riD. In fact, it may happen that riC and riD are disjoint, and such cases are not so pathological as one might think. For example, takeC := 0⊕[0,1]and D:={(λ, µ)∈R2 :λ, µ∈[0,1]}. That is, D is a square on R2, andC is its bottom edge. In this case we haveC ⊆D. Yet, we have seen in the previous paragraph that riC= 0⊕(0,1). Moreover, affD=R2, which implies thatriD= intD={(λ, µ)∈R2:λ, µ∈(0,1)}, that is, riC andriDare disjoint in this case.
Still, there are some tools which allows us to work with relative interiors without much trouble.
The next theorem states, for a convex set C⊆E, that the segment between a point inriC with any other pointx¯∈clC is almost entirely contained in riC, with the only exception being the point x¯ itself.
Theorem 3.2.1 ([59, Theorem 6.1]). Let C ⊆ E be a nonempty convex set, let ˚x ∈ riC, and let x¯∈clC. Then (1−λ)˚x+λ¯x∈riX for every λ∈[0,1).
Even though the above theorem gives us some intuition about points in the relative interior, it is of no help if we want to show that a point from a set lies in its relative interior. The next theorem helps us in this sense, showing a very enlightening characterization of points in the relative interior of convex sets: a point˚x in a convex set C ⊆E is in its relative interior if and only if, for every x∈C, the line segment betweenx and˚x can be slightly extended in the direction of˚x. To obtain a grasp of why this holds, recall that a point˚x∈C is inriC if and only if there isε >0 such that (˚x+εB)∩affC⊆C whereB:={x∈E:hx, xi ≤1}. That is, there is a ball confined in the affine hull ofC and centered in˚x which lies entirely in C. Since˚x−x∈affC for anyx∈C, one can at
least intuitively see that if the segment between˚x and every point x∈C can be extended in the direction of˚x, there must be such a “ball confined inaffC” from the definition of relative interior.
Theorem 3.2.2 ([59, Theorem 6.4]). Let C⊆Ebe a nonempty convex set and let˚x∈C. Then
˚x∈riC if and only if for each x∈C there isµ >1 such that(1−µ)x+µ˚x∈C .
As an application of the above theorem, let us compute the relative interior of some kinds of polyhedra.
Corollary 3.2.3. Let A∈Rm×n,b∈Rm,C ∈Rk×n, and d∈Rk be such that there is˚x ∈P :=
{x∈Rn:Ax≤b, Cx=d} such thatA˚x < b. Then riP ={x∈Rn:Ax < b, Cx=d}.
Proof. DefineP0 :={x∈Rn:Ax < b, Cx=d} and letx¯∈P. For anyµ∈R\ {0} andx∈P we have C((1−µ)x+µ¯x) =d. Thus, by Theorem 3.2.2 it suffices to show that for everyx∈P there is µ >1such thatA((1−µ)x+µ¯x)< b if and only ifA¯x < b (i.e.x¯∈P0).
First, suppose that for any x∈P there isµ >1such that A((1−µ)x+µ¯x)< b. In particular, there isµ >¯ 1such thatz:= (1−µ)˚¯ x+ ¯µ¯x∈P. Since1−µ <¯ 0 we have
Az=A((1−µ)˚¯ x+ ¯µ¯x) = (1−µ)A˚¯ x+ ¯µA¯x >(1−µ)b¯ + ¯µA¯x.
Sincez∈P, we haveAz≤b, which holds if and only if µb > µA¯x, that is, if and only ifAx < b.¯ Suppose now that Ax < b, and let¯ x∈P. Setr :=b−A¯x >0 and s:=b−Ax≥0. Note that, for any µ >1, by settingxµ:= (1−µ)x+µ¯x we have
Axµ=A((1−µ)x+µ¯x) =Ax+µ(A¯x−Ax) =b−s+µ(s−r). (3.1) If s−r ≤0, then we are done since, in this case, Axµ≤b−s≤b for any µ >1. Thus, suppose there isi∈[n]withsi−ri >0 and set
¯
µ:= min si
si−ri
:i∈[n], si−ri >0
>1,
and let i∗ ∈ [n] such that si∗(si∗ −ri∗)−1 attains the above minimum. It only remains to show that Axµ¯ ≤b. Let i ∈[n]. If si−ri ≤ 0, then by (3.1) we have (A¯x)i ≤bi. On the other hand, if si−ri >0, then
(Axµ¯)i=bi−si+ ¯µ(si−ri) =bi−si+ si∗ si∗−ri∗
(si−ri)≤bi−si+ si si−ri
(si−ri) =bi. Let us look at the idea of lower semi-continuity for functions which, maybe surprisingly, is just the translation of the closure property of the epigraph. Let f:E → (−∞,+∞]. The function f is lower semi-continuous atx ∈ Eif f(x) = lim infy→xf(y). One may note that a continuous function is, in particular, lower semi-continuous. The next theorem shows that for a functionf to be lower semi-continuous is equivalent to its epigraphepif being closed.
Theorem 3.2.4 ([59, Theorem 7.1]). Let f:E→(−∞,+∞]. The following are equivalent:
(i) f is lower semi-continuous onE;
(ii) For everyα∈Rthe set{x∈E:f(x)≤α} is closed;
(iii) The epigraph off is a closed set inE⊕R.
The above theorem makes it natural to define the closure of a function f as the function whose epigraph iscl(epif). Formally, theclosure off:E→(−∞,+∞] is the functionclf:E→ (−∞,+∞]given by
(clf)(x) := inf{µ∈R:x⊕µ∈cl(epif)}, ∀x∈E.
That is, clf for some functionf is the function whose epigraph is cl(epif). If f:E→[−∞,+∞] is such that there isx∈Ewithf(x) =−∞, we set(clf)(x) :=−∞for everyx∈E. Moreover, we say thatf:E→[−∞,+∞]isclosedif clf =f. The next theorem shows that the closure operation for convex functions yields closed convex functions and does not change the functions by much. Namely, a function and its closure differ maybe only on the border of the domain.
Theorem 3.2.5 ([59, Theorem 7.4]). Let f:E→ (−∞,+∞]be a proper convex function. Then clf is a proper closed convex function, andf(x) = (clf)(x) for every x∈ri(domf).
With the above theorem we know where a function and its closure can differ. Yet, we have not shown, besides taking the inferior limit, a way to discover the value of the closure of the function at one of these boundary points. The next theorem shows a simpler way to obtain the values of the closure of a function.
Theorem 3.2.6 ([59, Theorem 7.5]). Let f:E→(−∞,+∞]be a proper convex function and let
˚x∈ri(domf). Then,
(clf)(x) = lim
λ↑1f(λx+ (1−λ)˚x), ∀x∈E.
Finally, in the same way that the sum of continuous functions is continuous, we would like the sum of closed convex functions to be a closed convex function as well. The next theorem states exactly this.
Theorem 3.2.7 ([59, Theorem 9.3]). Letf1, . . . , fm:E→(−∞,+∞]be convex. Iffi is closed for each i∈[m], thenPm
i=1fi is a closed convex function.