Two Extensions to Finsler's Recurring Theorem |
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Authors: | C. Hamburger |
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Affiliation: | (1) Hohle Gasse 77, D-53177 Bonn, Germany , DE |
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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 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 . Accepted 22 February 1998 |
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Keywords: | . Recurring theorem Quadratic forms Lower semicontinuity Polyconvexity Rank-one convexity. AMS Classification. 15A63 15A48 49J45. |
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