Appendix

IfE is a topological vector space over R (with the Hausdorff property). Then the affine hull P of any two distinct points v1, w1 of E is a well defined 1- dimensional subset of E. Indeed we can define a homeomorphism between P

and R in the following way: for all xPR, let fpxq “xw1` p1´xqv1, so that the image off is indeed the affine hull of v andw, and it is clearly one-to-one with fp0q “ v1 and fp1q “ w1. Moreover, if we define the order ďf such that

for all x, y PR we have

xďy ô fpxq ďf fpyq

thenf is an order isomorphism betweenpR;ďqandpP;ďfq, and as such every

subset of P has a least upper bound (lub) and a greatest lower bound (glb) underďf.

Let Z be a compact, convex subset of E with cardinality greater than two. Define Qu :“ tPα :“ txα` p1´xqu: xP Ru :α P Zu to be the collection of

affine hulls through pairs of points ofZ, such that for every pair, one element of the pair is the point u. We define u to be an extremal point of Z if for every Pα in Qu, u is the glb of Pα XZ. Thus, by definition, if u is an ex-

tremal point ofZ, then for an arbitrary element Pα of Qu with f :RÝÑ Pα,

fpxq:“xα` p1´xqu, then uďf v for all v inPαXZ.

Claim 3.5. Ifuis an extremal point of the compact, convex set with cardinality greater than two, Z, then it lies in the boundary of Z relative to its affine hull (henceforth we refer to this as its relative boundary). Moreover, for every

P P Qu, the lub ofPXZ lies in the relative boundary ofZ, andPXZ is order

homeomorphic to a compact interval in R.

Proof. Suppose not. Take an arbitrary element P of Qu generated by u and

α. Then either u P PzZ or u P W :“ P XripZq, where rip¨q denotes the interior of a set relative to its affine hull (its relative interior). Since both P

and Z are closed subsets of E, P XZ is a closed subset of Z. This implies thatP XZ is compact (Munkres 2000 p181). Then by the extreme value the- orem (Munkres 2000 p190) there exists a pointsv1 and w1 inP XZ such that

f´1

pv1q ď f´1

pu1q ď f´1

pw1q for every u1 P P XZ. This implies that v1 ďf u1

for every u1 P P XZ, so that v1 “ u and u is therefore an element of P XZ. Thereforeu is an element of W.

The extreme value theorem also implies that P XZ contains its lub which we denote by v. Since both P and Z are convex, so is P XZ. Thus P XZ

is a compact convex set, and it therefore contains every point u1 such that

uďf u1 ďf v lies in P XZ. This implies that 1“f´1pαq ď f´1pvq “:y, and

for allu1 PP XZ, 0“ f´1

puq ăf´1

pu1q ďf´1

pvq “y. In other words, there exists a compact intervalr0, ysthat is homeomorphic to P XZ.

If v P W :“ ripZq XP, since ripZq is open in Z and P XZ is a subspace of P, W is open in P and strictly contained in P XZ. Thus f´1pWq is open in R and strictly contained in r0, ys. If z is the supremum of f´1

pWq, then z cannot be contained in f´1

pWq, for there would exist ǫą0 such that

z`ǫ P f´1

pWq and z ă z `ǫ. Thus z R f´1

pWq, so that x ă z for every

x P f´1

pWq; however if this true y ă z so that z R P XZ. The same proof shows thatu lies in the relative boundary ofZ.

We now make a suitable extension of this result to arbitrary closed subsets of

Z.

Claim 3.6. Let E, Z, Qu and u be defined as above. If K is a closed subset

of Z then every element α of K lies in a set P XZ for some P P Qu, and

the extremal pointu is a lower bound for the set P XK. Moreover, the lub of

P XK is contained in the relative boundary ofK.

Proof. The definition of Qu is such that every point in Z and therefore a

fortiriori K, is contained in the intersection of Z with some affine hullP P Q. Fix a point α P K. As K is a closed subset of Z, and the latter is compact,

K is also compact. Similarly, P XK is compact and by the extreme value theorem, it contains both its glb and its lub, and we denote these by v and w, respectively. Since v P P XK Ă P XZ, u ďf v and thus u is a lower bound

forPXK. Then by an identical argument to the preceding claim, we see that

wis an element of the relative boundary of K.

Claim 3.7. If suppPXKq ‰suppPXZq, then the least upper bound ofPXK

is an element of bdZK, the boundary of K relative to Z.

The intuition behind this claim is that the relative boundary of K is not in general equal to the boundary of K relative to Z. The case where they are not equal is whereK intersects the relative boundary of Z, so that part of the relative boundary of K is contained in the interior of K in the topology ofZ.

Proof. If k :“suppP XKq ‰ z :“ suppP XZq, then k ăf z as K is a subset

of Z. By claims (3.5) and (3.6), k is an element of the closed set P XK and

z P pPzKq XZ “ pP XZqzKq. Since P XZ is isomorphic to r0, ys for some 0ă y ă 8, with f´1 pzq “y, if we define x :“ f´1 pkq, then px, ys is open in r0, ysand contained inW :“f´1 ppPXZqzKq. Moreover, sincexP f´1 pPXKq,

xseparates px, ys fromWzpx, ys. Then by continuity of f´1

and the fact that connectedness is a topological property, fprx, ysq “ clPfppx, ysq and it is a

closed component of clPppP XZqzKq. Thus k is an element of the set

`

pP XZq XK˘XclPppP XZqzKqq Ă

`

ZXK˘XclZpZzKqq (3.9)