1540 CHAPTER 48. MULTIFUNCTIONS AND THEIR MEASURABILITY

48.2 Existence of Measurable Fixed Points48.2.1 Simplices And LabelingFirst define an n simplex, denoted by [x0, · · · ,xn], to be the convex hull of the n+1 points,{x0, · · · ,xn} where {xi−x0}n

i=1 are independent. Thus

[x0, · · · ,xn]≡

{n

∑i=0

tixi :n

∑i=0

ti = 1, ti ≥ 0

}.

Since {xi−x0}ni=1 is independent, the ti are uniquely determined. If two of them are

n

∑i=0

tixi =n

∑i=0

sixi

Thenn

∑i=0

ti (xi−x0) =n

∑i=0

si (xi−x0)

so ti = si for i ≥ 1. Since the si and ti sum to 1, it follows that also s0 = t0. If n ≤ 2,the simplex is a triangle, line segment, or point. If n ≤ 3, it is a tetrahedron, triangle, linesegment or point. To say that {xi−x0}n

i=1 are independent is to say that {xi−xr}i̸=r areindependent for each fixed r. Indeed, if xi−xr = ∑ j ̸=i,r c j (x j−xr) , then you would have

xi−x0 +x0−xr = ∑j ̸=i,r

c j (x j−x0)+

(∑

j ̸=i,rc j

)x0

and it follows that xi − x0 is a linear combination of the x j − x0 for j ̸= i, contrary toassumption.

A simplex S can be triangulated. This means it is the union of smaller sub-simplicessuch that if S1,S2 are two simplices in the triangulation, with

S1 ≡[z1

0, · · · ,z1m], S2 ≡

[z2

0, · · · ,z2p]

thenS1∩S2 =

[xk0 , · · · ,xkr

]where

[xk0 , · · · ,xkr

]is in the triangulation and{

xk0 , · · · ,xkr

}={

z10, · · · ,z1

m}∩{

z20, · · · ,z2

p}

or else the two simplices do not intersect. Does there exist a triangulation in which allsub-simplices have diameter less than ε? This is obvious if n ≤ 2. Supposing it to be truefor n− 1, is it also so for n? The barycenter b of a simplex [x0, · · · ,xn] is just 1

1+n ∑i xi.This point is not in the convex hull of any of the faces, those simplices of the form[x0, · · · , x̂k, · · · ,xn] where the hat indicates xk has been left out. Thus [x0, · · · ,b, · · · ,xn]is a n simplex also. Now in general, if you have an n simplex [x0, · · · ,xn] , its diameter is