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Invariant measure under the affine group over $$\mathbb Z$$. (English) Zbl 1298.52016
In this paper, the author studies invariants for equidissectability of rational polyhedra (finite unions of rational simplices) under piecewise affine unimodular transformations, that is, mappings in $$\mathcal{G}_n = \mathrm{GL}(n,\mathbb{Z}) \ltimes \mathbb{Z}^n$$. He constructs a family $$\lambda_0,\ldots,\lambda_n$$ of functions, of which $$\lambda_d$$ is called $$d$$-dimensional rational measure, which are exactly characterized by six properties – invariance under $$\mathcal{G}_n$$, valuation, conservatism under embedding in higher dimensions, inductive calculation for pyramids, normalization (by unit cubes) and proportionality to corresponding Lebesgue measure in fixed subspaces.

MSC:
 52B45 Dissections and valuations (Hilbert’s third problem, etc.) 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) 52A38 Length, area, volume and convex sets (aspects of convex geometry) 51M25 Length, area and volume in real or complex geometry 20H05 Unimodular groups, congruence subgroups (group-theoretic aspects) 20H25 Other matrix groups over rings
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