On simple polytopes. (English) Zbl 0803.52007

The author studies the subalgebra \(\Pi (P)\) of the polytope algebra \(\Pi\) generated by the classes of summands of a simple polytope \(P\) in Euclidean \(d\)-space. The most important result is that the weight space \(\Xi_ r (P)\) has dimension \(h_ r (P)\) for \(r \in \{0, \dots, d\}\), where \((h_ 0(P), \dots, h_ d(P))\) is the \(h\)-vector of \(P\). Moreover, the paper gives a new proof of the necessity of McMullen’s conditions in the well known \(g\)-theorem describing the possible \(f\)-vectors of simple \(d\)-polytopes. The present proof goes entirely within convexity. The aim of the author was to avoid deep techniques of algebraic geometry like in the proof by R. P. Stanley [Adv. Math. 35, 236-238 (1980; Zbl 0427.52006)].


52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)


Zbl 0427.52006
Full Text: DOI EuDML


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