Two Extensions to Finsler's Recurring Theorem

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 .