MathDBBV-alpha positivity conjecture for regular permutohedra
BV-alpha positivity conjecture for regular permutohedra
Let be a polytope. In McMullen's formula, write the local contribution associated with a face as , and use the Berline–Vergne construction for these local terms, called the BV--valuation. The polytope is BV--positive when all BV- values associated with are positive.
BV-alpha positivity conjecture. Every regular permutohedron is BV--positive.
The conjecture is based on computations in small dimensions, where the symmetry of regular permutohedra makes the recursive BV--valuation tractable. A general proof is not supplied and the conjecture remains open.
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Equivalent formulations 1
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
BV--positivity conjecture for regular permutohedra
Let denote the regular permutohedron, the usual permutohedron associated with . The BV--valuation is the valuation used in the paper's McMullen-type formula, and a polytope is BV--positive when the corresponding coefficients in that formula are positive. BV--positivity conjecture. Every regular permutohedron is BV--positive. This conjecture is a reduction of the Ehrhart positivity conjecture for integral generalized permutohedra, and the paper reports partial progress toward both statements.
source: Fu Liu, “On positivity of Ehrhart polynomials”, arXiv:1711.09962 (2018).
Sources & referencesView supporting material
Primary source
Federico Castillo and Fu Liu, “Berline-Vergne valuation and generalized permutohedra”, arXiv:1509.07884 (2017).
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