Two Extensions to Finsler's Recurring Theorem

Two Extensions to Finsler's Recurring Theorem Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R n : (i) f( ξ ) >0 for all ξ≠ 0 with g( ξ ) =0; (ii) f-λ g>0 for some λ∈ R. We prove two extensions: 1. We admit a vector-valued quadratic form g: R n → R k , for which we show that (i) implies that f-λ . . . g>0 on an ( n-k+1 ) -dimensional subspace Y $\subset$ R n for some λ∈ R k . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R n by a real vector space X . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Two Extensions to Finsler's Recurring Theorem

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Publisher
Springer-Verlag
Copyright
Copyright © Inc. by 1999 Springer-Verlag New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002459900121
Publisher site
See Article on Publisher Site

Abstract

Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R n : (i) f( ξ ) >0 for all ξ≠ 0 with g( ξ ) =0; (ii) f-λ g>0 for some λ∈ R. We prove two extensions: 1. We admit a vector-valued quadratic form g: R n → R k , for which we show that (i) implies that f-λ . . . g>0 on an ( n-k+1 ) -dimensional subspace Y $\subset$ R n for some λ∈ R k . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R n by a real vector space X .

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 2074

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