# An introduction to Reshetnyak's theory of subharmonic distances

@inproceedings{Fillastre2020AnIT, title={An introduction to Reshetnyak's theory of subharmonic distances}, author={Franccois Fillastre}, year={2020} }

A distance may be defined on a domain of the plane, by considering a suitable set of admissible curves, and a function λ such that its square root is integrable along any admissible curve. More precisely, to integrate the square root of λ gives a notion of length of the curve, and taking the infimum of these lengths between two points endows the domain with a pseudo-distance. If λ is C∞, one obtains a Riemannian distance. Other basic famous examples are flat metrics with conical singularities…

## 3 Citations

Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

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We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as…

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We prove an inequality of Bonnesen type for the real projective plane, generalizing Pu’s systolic inequality for positively-curved metrics. The remainder term in the inequality, analogous to that in…

Triangulating metric surfaces

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We prove that any length metric space homeomorphic to a surface may be decomposed into non-overlapping convex triangles of arbitrarily small diameter. This generalizes a previous result of…

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