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An analytical approach to the rational simplex problem. (English) Zbl 1299.51010
Summary: In 1973, Jeff Cheeger and James Simons raised the following question that still remains open and is known as the rational simplex problem: given a geodesic simplex in spherical 3-space so that all of its interior dihedral angles are rational multiples of \(\pi\), is it true that its volume is a rational multiple of the volume of the 3-sphere? An analytical approach to the rational simplex problem is proposed by deriving a function \(f(t)\), defined as an integral of an elementary function, such that if there is a rational \(t\), close enough to zero, such that the value \(f(t)\) is an irrational number then the answer to the rational simplex problem is negative.
MSC:
51M16 Inequalities and extremum problems in real or complex geometry
51M25 Length, area and volume in real or complex geometry
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