The gradient maps associated to certain non-homogeneous cones. (English) Zbl 1086.52501
Summary: The gradient map associated to a regular open convex cone gives a diffeomorphism from the cone onto its dual cone. If the cone is homogeneous, the inverse of the map is known to be equal to the gradient map associated to the dual cone. However, we show that this is no longer true for a general case by presenting a simple counterexample.
MSC:
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 53A15 | Affine differential geometry |
| 52A15 | Convex sets in \(3\) dimensions (including convex surfaces) |
References:
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| [2] | J. Faraut and A. Korányi, Analysis on symmetric cones , Oxford Univ. Press, New York, 1994. · Zbl 0841.43002 |
| [3] | T. Nomura, Family of Cayley transforms of a homogeneous Siegel domain parametrized by admissible linear forms, Differential Geom. Appl. 18 (2003), no. 1, 55-78. · Zbl 1023.22007 · doi:10.1016/S0926-2245(02)00098-0 |
| [4] | È. B. Vinberg, The theory of homogeneous convex cones, Trans. Moscow Math. Soc. 12 (1963), 340-403. · Zbl 0138.43301 |
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