# Euclidean distance cone faces

(Difference between revisions)
 Revision as of 20:21, 26 June 2008 (edit)m (Euclidean distance cone moved to Euclidean distance cone faces: This is an open question.)← Previous diff Current revision (16:42, 11 November 2009) (edit) (undo)m (Protected "Euclidean distance cone faces" [edit=autoconfirmed:move=autoconfirmed]) (8 intermediate revisions not shown.) Line 1: Line 1: The question remains open whether all faces of the cone of Euclidean distance matrices [itex]\,\mathbb{EDM}^N\![/itex] The question remains open whether all faces of the cone of Euclidean distance matrices [itex]\,\mathbb{EDM}^N\![/itex] - '''('''whose dimension is less than the dimension of the cone''')''' + + '''('''whose dimension is less than dimension of the cone''')''' + are exposed like they are for the positive semidefinite cone. are exposed like they are for the positive semidefinite cone. + + For a better explanation, see section 6.5.3 in [http://meboo.convexoptimization.com/BOOK/ConeDistanceMatrices.pdf Cone of Distance Matrices]. + + Definition of ''exposure'' is in [http://meboo.convexoptimization.com/BOOK/convexgeometry.pdf Convex Geometry]. + + Basically, the question asks whether all faces of [itex]\,\mathbb{EDM}^N\![/itex] can be defined by intersection with a supporting hyperplane; that intersection is termed ''exposure.''

## Current revision

The question remains open whether all faces of the cone of Euclidean distance matrices $LaTeX: \,\mathbb{EDM}^N\!$

(whose dimension is less than dimension of the cone)

are exposed like they are for the positive semidefinite cone.

For a better explanation, see section 6.5.3 in Cone of Distance Matrices.

Definition of exposure is in Convex Geometry.

Basically, the question asks whether all faces of $LaTeX: \,\mathbb{EDM}^N\!$ can be defined by intersection with a supporting hyperplane; that intersection is termed exposure.