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Geometry of the cone of positive quadratic forms
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Peter M. Gruber
Published/Copyright:
January 30, 2009
Abstract
Let a quadratic form on 𝔼d be represented by its coefficient vector in 𝔼(1/2)d(d+1). Then, to the family of all positive semidefinite quadratic forms on 𝔼d there corresponds a closed convex cone 𝒬d in 𝔼(1/2)d(d+1) with apex at the origin. We describe its exposed faces and show that the families of its extreme and exposed faces coincide. Using these results, flag transitivity, neighborliness, singularity and duality properties of 𝒬d are shown. The isometries of the cone 𝒬d are characterized and we state a conjecture describing its linear symmetries. While the cone 𝒬d is far from being polyhedral, the results obtained show that it shares many properties with highly symmetric, neighborly and self dual polyhedral convex cones.
Received: 2007-06-04
Published Online: 2009-01-30
Published in Print: 2009-January
© de Gruyter 2009
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Articles in the same Issue
- Central extensions of rank 2 groups and applications
- Generating abelian groups by addition only
- Intrinsic ultracontractivity for non-symmetric Lévy processes
- K-Theory of non-linear projective toric varieties
- Wakamatsu tilting modules with finite FP-injective dimension
- The catenary and tame degree of numerical monoids
- On extending Prüfer rings in central simple algebras
- Geometry of the cone of positive quadratic forms
