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A cone ''C'' in a vector space ''X'' is said to be ''self-dual'' if ''X'' can be equipped with an [[inner product]] ⟨⋅,⋅⟩ such that the internal dual cone relative to this inner product is equal to ''C''.<ref>Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.</ref>
Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.
This is slightly different
For instance, the above definition makes a cone in '''R'''<sup>''n''</sup> with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in '''R'''<sup>''n''</sup> is equal to its internal dual.
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