Distance optimization and the extremal variety of the Grassmann variety. (English) Zbl 1342.51010

Leventides, John; Petroulakis, George; Karcanias, Nicos

J. Optim. Theory Appl. 169, No. 1, 1-16 (2016).

Summary: The approximation of a multivector by a decomposable one is a distance-optimization problem between the multivector and the Grassmann variety of lines in a projective space. When the multivector diverges from the Grassmann variety, then the approximate solution sought is the worst possible. In this paper, it is shown that the worst solution of this problem is achieved, when the eigenvalues of the matrix representation of a related two-vector are all equal. Then, all these pathological points form a projective variety. We derive the equation describing this projective variety, as well as its maximum distance from the corresponding Grassmann variety. Several geometric and algebraic properties of this extremal variety are examined, providing a new aspect for the Grassmann varieties and the respective projective spaces.

MSC:

51K05	General theory of distance geometry
57Q55	Approximations in PL-topology
08B30	Injectives, projectives
11E25	Sums of squares and representations by other particular quadratic forms

Keywords:

distance geometry problems; optimization; approximations; projective varieties; sums of squares and representations

Software:

Sostools; Macaulay2; Matlab

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