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By Frithjof Nevanlinna, Rolf Nevanlinna

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Show that u = P x, v = Q x are linear transformations with the property Pax=Px, Qsx=Qx. P x is called the projection of x onto U in the "direction" I' and Q x = v the projection of x onto V in the direction U. 8. Show conversely : If P x is a linear transformation of Rs with the property P2 X. = P x, there exist uniquely determined linearly independent complements U and V of R°' such that u = P x is the projection of x onto U in the direction V and v = x - P x = Q x is the projection of x onto V in the direction U.

D (xi - xo) ks ... k.. For a fixed ro, 1)o is here a nonzero (in - 1)-linear alternating function . . It,,, of the side simplex s" which can be used as the fundamental form of the space Ut -'. of the edges It ! 4. Simplicial subdivisions. e. the closed convex hull of the points x0, ... , x,"). e. a finite set of closed tit-dimensional subsimplexes s;" (i -= 1, 2, ... N) with the properties: 1. The sinsplex s' is the union of the subsitnplexes s,". 2. " and s"I hare no common interior points.

K (:5 2'" - 1), we will prove their validity for i=k±1. By, assumption sk, any two of which have no common interior point, and the face sk'-' separates Pk from the complement Qk. As we have seen, the face sk'' also separates the simplexes s. and sty,, and therefore Hence, s', has no interior points in common with any of the simplexes s;', ... , s,t . The union P,t u sk f, is a polyhedron cut by the face sk;, from the given simplex s"and having the point x0 as a vertex. , c Qk . 00 Thus, P,t u s* , = Pk'*,, and PR+, is, as stated, the union of the simplexes sw , , plexes .

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