# 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{sup n} : (i) f( {xi} ) >0 for all {xi}{ne} 0 with g( {xi} ) =0; (ii) f-{lambda} g>0 for some {lambda} element of R. We prove two extensions: 1. We admit a vector-valued quadratic form g:R{sup n{yields}}R{sup k} , for which we show that (i) implies that f-{lambda} . . . g>0 on an ( n-k+1)-dimensional subspace Y is subset of R{sup n} for some {lambda} element of R{sup k} . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R{sup n} by a real vector space X.

- Authors:

- Hohle Gasse 77, D-53177 Bonn (Germany)

- Publication Date:

- OSTI Identifier:
- 21067543

- Resource Type:
- Journal Article

- Journal Name:
- Applied Mathematics and Optimization

- Additional Journal Information:
- Journal Volume: 40; Journal Issue: 2; Other Information: DOI: 10.1007/s002459900121; Copyright (c) 1999 Springer-Verlag New York Inc.; Article Copyright (c) Inc. 1999 Springer-Verlag New York; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0095-4616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FUNCTIONS; MANY-DIMENSIONAL CALCULATIONS; MATHEMATICAL LOGIC; MATHEMATICAL SPACE; VECTORS

### Citation Formats

```
Hamburger, C.
```*Two Extensions to Finsler's Recurring Theorem*. United States: N. p., 1999.
Web. doi:10.1007/S002459900121.

```
Hamburger, C.
```*Two Extensions to Finsler's Recurring Theorem*. United States. doi:10.1007/S002459900121.

```
Hamburger, C. Wed .
"Two Extensions to Finsler's Recurring Theorem". United States. doi:10.1007/S002459900121.
```

```
@article{osti_21067543,
```

title = {Two Extensions to Finsler's Recurring Theorem},

author = {Hamburger, C.},

abstractNote = {Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R{sup n} : (i) f( {xi} ) >0 for all {xi}{ne} 0 with g( {xi} ) =0; (ii) f-{lambda} g>0 for some {lambda} element of R. We prove two extensions: 1. We admit a vector-valued quadratic form g:R{sup n{yields}}R{sup k} , for which we show that (i) implies that f-{lambda} . . . g>0 on an ( n-k+1)-dimensional subspace Y is subset of R{sup n} for some {lambda} element of R{sup k} . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R{sup n} by a real vector space X.},

doi = {10.1007/S002459900121},

journal = {Applied Mathematics and Optimization},

issn = {0095-4616},

number = 2,

volume = 40,

place = {United States},

year = {1999},

month = {9}

}

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