Then you get ∑_{j≠m}c_{j} = 0 and each c_{j} = 0 for j≠m,0. Thus c_{0} = 0 also because the sum is 0 and all other
c_{j} = 0.

Since

{xi − x0}

_{i=1}^{n} is independent, the t_{i} are uniquely determined. If two of them are

n∑ ∑n
tixi = sixi
i=0 i=0

Then

∑n ∑n
ti(xi − x0) = si(xi − x0)
i=0 i=0

so t_{i} = s_{i} for i ≥ 1 by independence. Since the s_{i} and t_{i} sum to 1, it follows that also s_{0} = t_{0}. If n ≤ 2, the
simplex is a triangle, line segment, or point. If n ≤ 3, it is a tetrahedron, triangle, line segment or
point.

A simplex S can be triangulated. This means it is the union of smaller sub-simplices such that if S_{1},S_{2}
are two simplices in the triangulation, with

S ≡ [z1,⋅⋅⋅,z1 ], S ≡ [z2,⋅⋅⋅,z2]
1 0 m 2 0 p

then

S1 ∩S2 = [xk0,⋅⋅⋅,xkr]

where

[x ,⋅⋅⋅,x ]
k0 kr

is in the triangulation and

{xk0,⋅⋅⋅,xkr} = {z10,⋅⋅⋅,z1m} ∩{z20,⋅⋅⋅,z2p}

or else the two simplices do not intersect.

Does there exist a triangulation in which all sub-simplices have diameter less than ε? This is
obvious if n ≤ 2. Supposing it to be true for n − 1, is it also so for n? The barycenter b of
a simplex

[x ,⋅⋅⋅,x ]
0 n

is just

-1-
1+n

∑_{i}x_{i}. This point is not in the convex hull of any of the
faces, those simplices of the form

[x ,⋅⋅⋅,ˆx ,⋅⋅⋅,x ]
0 k n

where the hat indicates x_{k} has been left
out. Thus

[x ,⋅⋅⋅,b,⋅⋅⋅,x ]
0 n

is a n simplex also. Now in general, if you have an n simplex

[x ,⋅⋅⋅,x ]
0 n

, its diameter is the maximum of

|x − x |
k l

for all k≠l. Consider

|b − x |
j

. It equals

||∑n 1 ||
| i=0n+1 (xi − xj)|

=

||∑ 1 ||
| i⁄=j n+1 (xi − xj)|

≤

n
n+1

diam

(S)

. Next consider the k^{th} face of S

[x0,⋅⋅⋅,ˆxk,⋅⋅⋅,xn]

. By induction, it has a triangulation into simplices which each have diameter no more
than

nn+1

diam

(S)

. Let these n − 1 simplices be denoted by

{ }
Sk1,⋅⋅⋅,Skmk

. Then the simplices

{[ ]}
Ski ,b

_{i=1,k=1}^{mk,n+1} are a triangulation of S such that diam

([ ])
Ski ,b

≤

nn+1

diam

(S)

. Do for

[Ski ,b ]

what was just done for S obtaining a triangulation of S as the union of what is obtained
such that each simplex has diameter no more than

(-n-)
n+1

^{2}diam

(S)

. Continuing this way
shows the existence of the desired triangulation. You simply do the process k times where