### Abstract

The Γ-limit for a sequence of length functionals associated with a one parameter family of Riemannian manifolds is computed analytically. The Riemannian manifold is of ‘two-phase’ type, that is, the metric coefficient takes values in {1, β}, with β sufficiently large. The metric coefficient takes the value β on squares, the size of which are controlled by a single parameter. We find a family of examples of limiting Finsler metrics that are piecewise affine with infinitely many lines of discontinuity. Such an example provides insight into how the limit metric behaves under variations of the underlying microscopic Riemannian geometry, with implications for attempts to compute such metrics numerically.

Original language | English |
---|---|

Pages (from-to) | 19-36 |

Number of pages | 18 |

Journal | Journal of Convex Analysis |

Volume | 22 |

Issue number | 1 |

Publication status | Published - 1 Jan 2015 |

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)

### Cite this

*Journal of Convex Analysis*,

*22*(1), 19-36.

**The Finsler Metric Obtained as the Γ-limit of a Generalised Manhattan Metric.** / Schwetlick, Hartmut; Sutton, Daniel C.; Zimmer, Johannes.

Research output: Contribution to journal › Article

*Journal of Convex Analysis*, vol. 22, no. 1, pp. 19-36.

}

TY - JOUR

T1 - The Finsler Metric Obtained as the Γ-limit of a Generalised Manhattan Metric

AU - Schwetlick, Hartmut

AU - Sutton, Daniel C.

AU - Zimmer, Johannes

PY - 2015/1/1

Y1 - 2015/1/1

N2 - The Γ-limit for a sequence of length functionals associated with a one parameter family of Riemannian manifolds is computed analytically. The Riemannian manifold is of ‘two-phase’ type, that is, the metric coefficient takes values in {1, β}, with β sufficiently large. The metric coefficient takes the value β on squares, the size of which are controlled by a single parameter. We find a family of examples of limiting Finsler metrics that are piecewise affine with infinitely many lines of discontinuity. Such an example provides insight into how the limit metric behaves under variations of the underlying microscopic Riemannian geometry, with implications for attempts to compute such metrics numerically.

AB - The Γ-limit for a sequence of length functionals associated with a one parameter family of Riemannian manifolds is computed analytically. The Riemannian manifold is of ‘two-phase’ type, that is, the metric coefficient takes values in {1, β}, with β sufficiently large. The metric coefficient takes the value β on squares, the size of which are controlled by a single parameter. We find a family of examples of limiting Finsler metrics that are piecewise affine with infinitely many lines of discontinuity. Such an example provides insight into how the limit metric behaves under variations of the underlying microscopic Riemannian geometry, with implications for attempts to compute such metrics numerically.

UR - http://www.scopus.com/inward/record.url?scp=85071450188&partnerID=8YFLogxK

M3 - Article

VL - 22

SP - 19

EP - 36

JO - Journal of Convex Analysis

JF - Journal of Convex Analysis

SN - 0944-6532

IS - 1

ER -