A New Look at Euler's Theorem for Polyhedra: A Comment

Abstract

In their interesting paper [6] authors, Branko Grunbaum and Geoffrey Shephard mention (page 125/126) . . . work of [20-25], which our note parallels to some extent. Nefns definition of polyhedral sets differs from ours, but is equivalent to it; his approach to Euler characteristic is same as ours. Actually, this parallelism concerns a major part of article and comes partly into conflict with my own work in field. The main reason is authors' inexplicable statement (p. 126) Nef [1978] defines as faces of a polyhedral set P any family of disjoint, relatively open convex sets that is a dissection of P. ... However, these 'faces' are, in general, not uniquely determined, and have only a limited geometric In fact my definition is one in [8, p. 6.2], or in [1, p. 98]. According to this definition faces of a polyhedron turn out to be relative interiors of faces in intuitive sense. They are uniquely determined and have a clear geometric significance. (Unfortunately, in [1] a typing error slipped in: On p. 98 in formula (06) U0 should be replaced by P n ao, and U by P n U). In a thorough discussion, several further points would have to be critically looked at. I confine myself to two of them: On page 122, Grunbaum and Shephard present their Theorem 2* concerning Euler characteristic of (not necessarily bounded) closed convex polyhedra. They overlook that I have published same result previously in [9, Satz 4, p. 68]. On page 117, authors define Euler characteristic X(P) (in same way as I have done in [10, Satz 2, pp. 44-45]): For a cell C (a nonempty relatively open convex polyhedron) we put x(C) = (-l)dimC, furthermore, X(0) = 0. If P Uc E c C represents P as a finite disjoint union of cells, then X(P) = Ec E c x(c) The definition is followed by three theorems, first of which asserts that X(P) does not depend on partition of P into cells. (For a proof of this Theorem see [10, Satz 2, pp. 44-45]). The second theorem states that X(P) = 1 if P is bounded, closed, and convex. The proofs are omitted since they follow usual techniques. From discovery of his theorem by Euler (in 1750) more than 200 years were needed to find a complete proof ([5, p. 134], [7, p. 94], [2]); for an extended discussion of historical development see [6, pp. 122-127]. It is therefore not evident what is meant by the usual techniques, at least in case of second theorem, which is an essential part of theorem of Euler-Schlafli (Leonhard Euler [3] and [4], and extension to higher dimensions in 1850 by Ludwig Schlafli [12]). In [11] I have published a particularly short and simple proof which I repeat here: First of all, every closed polyhedron P is disjoint union of its faces (without extP) [8, Satz 3, p. 6.7]. If P is closed and convex, faces are cells [8, Satz 8, p. 7.12]. Therefore X(P) = Lin=0( l)iti, where n = dim P and fi is number of faces of dimension i (with fn = 1 for relative interior of P). Now let P be bounded, closed, and convex, and let 4) denote family of all faces F of P with dim F < n. We choose an arbitrary point z E relintP. For each face F E ¢ we define F* = {x = z + A(F-z) with 0 < A < 1}, so F* is