Finding the convex envelope of a boundary datum using random geometric graphs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917355190222848 |
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| author | Deshayes, Aurelia Frevenza, Nicolás Miranda, Alfredo Rossi, Julio D. |
| author_facet | Deshayes, Aurelia Frevenza, Nicolás Miranda, Alfredo Rossi, Julio D. |
| contents | In this paper we approximate the convex envelope of a boundary datum inside a bounded domain in the Euclidean space. We work with a random graph that is obtained as random points with uniform distribution that are connected by proximity ($x\sim y$ when $|x-y|<r$). On the graph we solve an equation (that approximate the first eigenvalue of the Hessian of a smooth function) with an exterior datum. Under appropriate assumptions on $r$ we show that the unique solution to the equation in the graph converges to the convex envelope of the boundary datum as the number of points goes to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07696 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finding the convex envelope of a boundary datum using random geometric graphs Deshayes, Aurelia Frevenza, Nicolás Miranda, Alfredo Rossi, Julio D. Analysis of PDEs Probability 05C80, 60D05, 52A41, 35D40 In this paper we approximate the convex envelope of a boundary datum inside a bounded domain in the Euclidean space. We work with a random graph that is obtained as random points with uniform distribution that are connected by proximity ($x\sim y$ when $|x-y|<r$). On the graph we solve an equation (that approximate the first eigenvalue of the Hessian of a smooth function) with an exterior datum. Under appropriate assumptions on $r$ we show that the unique solution to the equation in the graph converges to the convex envelope of the boundary datum as the number of points goes to infinity. |
| title | Finding the convex envelope of a boundary datum using random geometric graphs |
| topic | Analysis of PDEs Probability 05C80, 60D05, 52A41, 35D40 |
| url | https://arxiv.org/abs/2602.07696 |