2 0 Ju n 20 02 Units , polyhedra , and a conjecture of Satake

ar X iv : h ep - p h / 02 06 17 1 v 2 2 4 Ju n 20 02 IPHE 2002 - 009 June 24 , 2002 B 0 – B 0 MIXING

ar X iv : h ep - p h / 02 06 27 0 v 1 2 6 Ju n 20 02 1 The charge asymmetry from Pomeron - Odderon interference in hard

Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions

Abstract. In 1996 Sabitov proved that the volume of an arbitrary simplicial polyhedron P in the 3-dimensional Euclidean space R satisfies a monic (with respect to V ) polynomial relation F (V, l) = 0, where l denotes the set of the squares of edge lengths of P . In 2011 the author proved the same assertion for polyhedra in R. In this paper, we prove that the same result is true in arbitrary dim...

ar X iv : 0 70 4 . 17 02 v 2 [ m at h . A G ] 2 6 Ju n 20 08 THE CANONICAL VOLUME OF 3 - FOLDS OF GENERAL TYPE WITH χ ≤ 0

We prove that the canonical volume K ≥ 1 30 for all 3-folds of general type with χ(O) ≤ 0. This bound is sharp.

ar X iv : 0 70 4 . 17 02 v 3 [ m at h . A G ] 2 7 Ju n 20 08 THE CANONICAL VOLUME OF 3 - FOLDS OF GENERAL TYPE WITH χ ≤ 0

We prove that the canonical volume K ≥ 1 30 for all 3-folds of general type with χ(O) ≤ 0. This bound is sharp.