31 is a particular case of a result proven in [43] and it has found various applications in shape classification [37] and in statistical analysis of data -see, e.g., [79, 78, 10, 44]. 284 CHAPTER 11, Theorem 11, 2008. ,
Surface Reconstruction by Voronoi Filtering, Discrete & Computational Geometry, vol.22, issue.4, pp.481-504, 1999. ,
DOI : 10.1007/PL00009475
URL : http://www.cs.ucdavis.edu/~amenta/pubs/3dcrust.ps.gz
Stability and Computation of Medial Axes - a State-of-the-Art Report, Math. Foundations of Scientific Visualization, Comp. Graphics, and Massive Data Exploration, pp.109-125, 2009. ,
DOI : 10.1007/b106657_6
URL : https://hal.archives-ouvertes.fr/hal-00468690
Weak witnesses for Delaunay triangulations of submanifolds, Proceedings of the 2007 ACM symposium on Solid and physical modeling , SPM '07, pp.143-150, 2007. ,
DOI : 10.1145/1236246.1236267
EFFICIENT DATA STRUCTURE FOR REPRESENTING AND SIMPLIFYING SIMPLICIAL COMPLEXES IN HIGH DIMENSIONS, International Journal of Computational Geometry & Applications, vol.24, issue.04, pp.279-304, 2012. ,
DOI : 10.1177/0278364909352700
URL : https://hal.archives-ouvertes.fr/hal-00579902
Power Diagrams: Properties, Algorithms and Applications, SIAM Journal on Computing, vol.16, issue.1, pp.78-96, 1987. ,
DOI : 10.1137/0216006
Voronoi Diagrams and Delaunay Triangulations, World Scientific, 2013. ,
DOI : 10.1142/8685
Géométrie (vols. 1-5). Fernand Nathan, 1977. ,
A weighted k-nearest neighbor density estimate for geometric inference, Electronic Journal of Statistics, vol.5, issue.0, pp.204-237, 2011. ,
DOI : 10.1214/11-EJS606
URL : https://hal.archives-ouvertes.fr/inria-00560623
Robust statistics , hypothesis testing, and confidence intervals for persistent homology on metric measure spaces. Found, Comp. Math, vol.14, issue.4, pp.745-789, 2014. ,
DOI : 10.1007/s10208-014-9201-4
URL : http://arxiv.org/pdf/1206.4581
Building Efficient and Compact Data Structures for Simplicial Complexes, Proc. 31st Symp, 2015. ,
DOI : 10.1145/253168.253192
URL : https://hal.archives-ouvertes.fr/hal-01145407
Triangulations in CGAL, Computational Geometry, vol.22, issue.1-3, pp.5-19, 2002. ,
DOI : 10.1016/S0925-7721(01)00054-2
URL : https://hal.archives-ouvertes.fr/hal-01179408
DELAUNAY STABILITY VIA PERTURBATIONS, International Journal of Computational Geometry & Applications, vol.24, issue.02, pp.125-152, 2014. ,
DOI : 10.1145/1667053.1667060
URL : https://hal.archives-ouvertes.fr/hal-00806107
THE STABILITY OF DELAUNAY TRIANGULATIONS, International Journal of Computational Geometry & Applications, vol.27, issue.5, pp.303-333, 2014. ,
DOI : 10.1007/s10711-008-9261-1
URL : https://hal.archives-ouvertes.fr/hal-00807050
A Probabilistic Approach to Reducing Algebraic Complexity of Delaunay Triangulations, Proc. 23rd European Symp. on Algorithms, pp.595-606, 2015. ,
DOI : 10.1007/s10711-008-9261-1
URL : https://hal.archives-ouvertes.fr/hal-01213070
Local criteria for triangulation of manifolds, Proc. 34st Symp, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-01661230
A coordinate system associated with points scattered on a surface, Computer-Aided Design, vol.36, issue.2, pp.161-174, 2004. ,
DOI : 10.1016/S0010-4485(03)00059-9
Manifold Reconstruction Using Tangential Delaunay Complexes, Discrete & Computational Geometry, vol.26, issue.1, pp.221-267, 2014. ,
DOI : 10.1137/S1064827502419154
URL : https://hal.archives-ouvertes.fr/hal-00487862
The reach, metric distortion, geodesic convexity and the variation of tangent spaces, Proc. 34st Symp, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-01661227
The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes, Algorithmica, vol.132, issue.23, pp.406-427, 2014. ,
DOI : 10.1063/1.3445267
URL : https://hal.archives-ouvertes.fr/hal-00707901
Bregman Voronoi Diagrams, Discrete & Computational Geometry, vol.12, issue.2, p.2010 ,
DOI : 10.1515/9781400873173
URL : https://hal.archives-ouvertes.fr/hal-00488441
Provably good sampling and meshing of surfaces, Graphical Models, vol.67, issue.5, pp.405-451, 2005. ,
DOI : 10.1016/j.gmod.2005.01.004
URL : https://hal.archives-ouvertes.fr/hal-00488829
Curved Voronoi Diagrams, Effective Computational Geometry for Curves and Surfaces, pp.67-116, 2006. ,
DOI : 10.1007/978-3-540-33259-6_2
URL : https://hal.archives-ouvertes.fr/hal-00488446
Anisotropic Delaunay Mesh Generation, SIAM Journal on Computing, vol.44, issue.2, pp.467-512, 2015. ,
DOI : 10.1137/140955446
URL : https://hal.archives-ouvertes.fr/inria-00615486
Algorithmic Geometry, 1998. ,
DOI : 10.1017/CBO9781139172998
Delaunay triangulation of a random sample of a good sample has linear size, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-01673170
Delaunay triangulation of manifolds. Found, Comp. Math, vol.18, issue.2, pp.399-431, 2018. ,
DOI : 10.1007/s10208-017-9344-1
URL : https://hal.archives-ouvertes.fr/hal-00879133
Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces, Probability Theory and Related Fields, vol.206, issue.1, pp.541-593, 2007. ,
DOI : 10.2140/pjm.1958.8.171
URL : https://hal.archives-ouvertes.fr/hal-00453883
Efficient and robust persistent homology for measures, Proc. 36th ACM-SIAM Symp. on Discrete Algorithms, pp.168-180, 2015. ,
DOI : 10.1137/1.9781611973730.13
URL : https://hal.archives-ouvertes.fr/hal-01074566
A course in metric geometry, Grad. Studies in Math, vol.33, 2001. ,
DOI : 10.1090/gsm/033
Deconvolution for the Wasserstein metric and geometric inference, Electronic Journal of Statistics, vol.5, issue.0, pp.1394-1423, 2011. ,
DOI : 10.1214/11-EJS646
URL : https://hal.archives-ouvertes.fr/inria-00607806
A simple triangulation method for smooth maniolds A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields, Bull. Amer. Math. Soc. J. ACM, vol.67, issue.421, pp.380-39067, 1961. ,
Semiconcave Functions, Hamilton- Jacobi Equations, and Optimal Control, Brikhauser, vol.58, 2004. ,
A geometric convection approach of 3d-reconstruction, 1st Symp. Geom. Processing, pp.218-229, 2003. ,
URL : https://hal.archives-ouvertes.fr/inria-00071898
Proximity of persistence modules and their diagrams, Proceedings of the 25th annual symposium on Computational geometry, SCG '09, pp.237-246, 2009. ,
DOI : 10.1145/1542362.1542407
URL : https://hal.archives-ouvertes.fr/inria-00292566
Gromov-Hausdorff Stable Signatures for Shapes using Persistence, Computer Graphics Forum, vol.33, issue.5, pp.1393-1403, 2009. ,
DOI : 10.1109/TPAMI.2006.208
URL : https://hal.archives-ouvertes.fr/hal-00772413
Normal cone approximation and offset shape isotopy, Computational Geometry, vol.42, issue.6-7, pp.6-7566, 2009. ,
DOI : 10.1016/j.comgeo.2008.12.002
URL : https://hal.archives-ouvertes.fr/inria-00124825
A Sampling Theory for Compact Sets in Euclidean Space, Discrete & Computational Geometry, vol.18, issue.3, pp.461-479, 2009. ,
DOI : 10.1007/s00454-009-9144-8
URL : https://hal.archives-ouvertes.fr/hal-00864493
Shape smoothing using double offsets, Proceedings of the 2007 ACM symposium on Solid and physical modeling , SPM '07, pp.183-192, 2007. ,
DOI : 10.1145/1236246.1236273
URL : https://hal.archives-ouvertes.fr/inria-00104582
Stability of Curvature Measures, Computer Graphics Forum, vol.26, issue.2, pp.1485-1496, 2009. ,
DOI : 10.1090/pspum/054.3/1216630
URL : https://hal.archives-ouvertes.fr/inria-00344903
Geometric inference for probability measures. Found, Comp. Math, vol.11, issue.6, pp.733-751, 2011. ,
DOI : 10.1007/s10208-011-9098-0
URL : https://hal.archives-ouvertes.fr/hal-00772444
Persistence stability for geometric complexes, Geometriae Dedicata, vol.33, issue.2, pp.193-214, 2014. ,
DOI : 10.1007/s00454-004-1146-y
URL : https://hal.archives-ouvertes.fr/hal-00923560
Subsampling methods for persistent homology, Proc. 32nd Int. Conf. on Machine Learning Conference Proceedings, pp.2143-2151, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-01073073
Robust topological inference: Distance to a measure and kernel distance. arXiv preprint arXiv, pp.1412-7197 ,
URL : https://hal.archives-ouvertes.fr/hal-01232217
The ?????-medial axis???, Graphical Models, vol.67, issue.4, pp.304-331, 2005. ,
DOI : 10.1016/j.gmod.2005.01.002
Weak feature size and persistent homology, Proceedings of the twenty-first annual symposium on Computational geometry , SCG '05, pp.255-262, 2005. ,
DOI : 10.1145/1064092.1064132
Stability and Computation of Topological Invariants of Solids in ${\Bbb R}^n$, Discrete & Computational Geometry, vol.37, issue.4, pp.601-617, 2007. ,
DOI : 10.1007/s00454-007-1309-8
Rates of convergence for robust geometric inference, Electronic Journal of Statistics, vol.10, issue.2, pp.2243-2286, 2016. ,
DOI : 10.1214/16-EJS1161
URL : https://hal.archives-ouvertes.fr/hal-01157551
Towards persistence-based reconstruction in euclidean spaces, Proceedings of the twenty-fourth annual symposium on Computational geometry , SCG '08, pp.232-241, 2008. ,
DOI : 10.1145/1377676.1377719
URL : https://hal.archives-ouvertes.fr/inria-00197543
The structure and stability of persistence modules, SpringerBriefs in Mathematics, 2016. ,
DOI : 10.1007/978-3-319-42545-0
URL : https://hal.archives-ouvertes.fr/hal-01330678
An optimal convex hull algorithm in any fixed dimension, Discrete & Computational Geometry, vol.16, issue.4, pp.377-409, 1993. ,
DOI : 10.1137/1116025
Critical points of distance functions and applications to geometry In Geometric Topology: recent developments, Montecani Terme, Lecture Notes in Math, vol.1504, pp.1-38, 1990. ,
On the curvature of piecewise flat spaces, Communications in Mathematical Physics, vol.12, issue.3, pp.405-454, 1984. ,
DOI : 10.1007/978-3-0348-5949-3
Manifold Reconstruction from Point Samples, Proc. 16th ACM-SIAM Symp. Discrete Algorithms, pp.1018-1027, 2005. ,
Optimization and Nonsmooth Analysis Applications of random sampling in computational geometry, II. Discrete Comp. Geom, vol.4, pp.387-421, 1983. ,
Stability of Persistence Diagrams, Discrete & Computational Geometry, vol.37, issue.1, pp.103-120, 2007. ,
DOI : 10.1007/s00454-006-1276-5
Computational Geom.: Algorithms and Applications, 2000. ,
A weak characterisation of the Delaunay triangulation, Geometriae Dedicata, vol.33, issue.2, pp.39-64, 2008. ,
DOI : 10.1007/978-1-4612-1098-6
Coverage in sensor networks via persistent homology, Algebraic & Geometric Topology, vol.10, issue.1, pp.339-358, 2007. ,
DOI : 10.1007/s00454-004-1146-y
Sur la sphère vide, Otdelenie Matematicheskii i Estestvennyka Nauk, pp.793-800, 1934. ,
Curve and Surface Reconstruction : Algorithms with Mathematical Analysis, 2007. ,
DOI : 10.1017/CBO9780511546860
URL : http://www.cis.ohio-state.edu/~tamaldey/paper/book/recon.ps.gz
Riemannian simplices and triangulations, Geometriae Dedicata, vol.41, issue.4, pp.91-138, 2015. ,
DOI : 10.1515/9781400877577
URL : http://doi.org/10.1007/s10711-015-0069-5
Algorithms in Combinatorial Geometry, EATCS Monographs on Theoretical Comp. Science, vol.10, 1987. ,
DOI : 10.1007/978-3-642-61568-9
The union of balls and its dual shape, Discrete & Computational Geometry, vol.133, issue.3-4, pp.415-440, 1995. ,
DOI : 10.1007/978-1-4612-4576-6
Geometry and Topology for Mesh Generation, 2001. ,
DOI : 10.1115/1.1445302
Surface reconstruction by wrapping finite point sets in space, pp.379-404, 2003. ,
DOI : 10.1007/978-3-642-55566-4_17
URL : http://www.cs.duke.edu/~edels/TriTop/Wrap.pdf
On the definition and the construction of pockets in macromolecules, Discrete Applied Mathematics, vol.88, issue.1-3, pp.83-102, 1998. ,
DOI : 10.1016/S0166-218X(98)00067-5
Computational topology: an introduction, 2010. ,
DOI : 10.1090/mbk/069
On the shape of a set of points in the plane, IEEE Transactions on Information Theory, vol.29, issue.4, pp.551-559, 1983. ,
DOI : 10.1109/TIT.1983.1056714
Topological Persistence and Simplification, Discrete & Computational Geometry, vol.28, issue.4, pp.511-533, 2002. ,
DOI : 10.1007/s00454-002-2885-2
URL : http://graphics.stanford.edu/~afra/papers/focs00/dcg.ps.gz
Smoothing and cleaning up slivers, Proceedings of the thirty-second annual ACM symposium on Theory of computing , STOC '00, pp.273-277, 2000. ,
DOI : 10.1145/335305.335338
URL : http://www.cs.iit.edu/~xli/paper/Conf/sliver-STOC00.pdf
Three-dimensional alpha shapes, ACM Transactions on Graphics, vol.13, issue.1, pp.43-72, 1994. ,
DOI : 10.1145/174462.156635
URL : http://arxiv.org/pdf/math/9410208v1.pdf
Topological data analysis with Bregman divergences, 33rd Symp. Comp. Geom, pp.1-39, 2017. ,
Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time, Proc. 21st Int. Symp. on Algorithms and Computation, pp.403-414, 2010. ,
DOI : 10.1007/s00373-007-0738-8
Confidence sets for persistence diagrams, The Annals of Statistics, vol.42, issue.6, pp.2301-2339, 2014. ,
DOI : 10.1214/14-AOS1252SUPP
URL : http://arxiv.org/pdf/1303.7117
Convergence rates for persistence diagram estimation in topological data analysis, J. Machine Learning Research, vol.16, pp.3603-3635, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-01073072
Optimal algorithms for approximate clustering, Proceedings of the twentieth annual ACM symposium on Theory of computing , STOC '88, pp.434-444, 1988. ,
DOI : 10.1145/62212.62255
Curvature measures, Transactions of the American Mathematical Society, vol.93, issue.3, pp.418-491, 1959. ,
DOI : 10.1090/S0002-9947-1959-0110078-1
Efficient simplicial reconstructions of manifolds from their samples, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.24, issue.10, 2002. ,
DOI : 10.1109/TPAMI.2002.1039206
URL : http://www.cs.rpi.edu/~freedd/publications/freedman_pami02.pdf
Size theory as a topological tool for computer vision, Pattern Recognition and Image Analysis, vol.9, pp.596-603, 1999. ,
Convergence of curvatures in secant approximations, Journal of Differential Geometry, vol.37, issue.1, pp.177-190, 1993. ,
DOI : 10.4310/jdg/1214453427
Tubular neighborhoods in Euclidean spaces. Duke Math, Journal, vol.52, issue.4, pp.1025-1046, 1985. ,
DOI : 10.1215/s0012-7094-85-05254-8
Algebraic Topology: a First Course, 1995. ,
DOI : 10.1007/978-1-4612-4180-5
Notes on Convex Sets, Polytopes, Polyhedra Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations, 2007. ,
DOI : 10.1007/978-1-4613-0137-0_9
URL : https://hal.archives-ouvertes.fr/inria-00193831
Riemannian Geometry, 1990. ,
URL : https://hal.archives-ouvertes.fr/hal-00002870
The flow complex: A data structure for geometric modeling, Proc. 14th ACM-SIAM Symp. Discrete Algorithms (SODA), pp.285-294, 2003. ,
DOI : 10.1016/j.comgeo.2007.01.002
URL : https://doi.org/10.1016/j.comgeo.2007.01.002
Shape Dimension and Intrinsic Metric from Samples of Manifolds, Discrete & Computational Geometry, vol.32, issue.2, pp.245-267, 2004. ,
DOI : 10.1007/s00454-004-1120-8
Clustering to minimize the maximum intercluster distance, Theoretical Computer Science, vol.38, issue.2-3, pp.293-306, 1985. ,
DOI : 10.1016/0304-3975(85)90224-5
Critical point theory for distance functions, Proceedings of Symposia in Pure Mathematics, 1993. ,
DOI : 10.1090/pspum/054.3/1216630
Witnessed k-Distance, Discrete & Computational Geometry, vol.40, issue.2, pp.22-45, 2013. ,
DOI : 10.1090/gsm/058
URL : https://hal.archives-ouvertes.fr/hal-00872490
Geometric approximation algorithms, 2011. ,
DOI : 10.1090/surv/173
URL : http://valis.cs.uiuc.edu/~sariel/teach/notes/aprx/book.pdf
Fast Construction of Nets in Low-Dimensional Metrics and Their Applications, SIAM Journal on Computing, vol.35, issue.5, pp.1148-1184, 2006. ,
DOI : 10.1137/S0097539704446281
Algebraic Topology, 2002. ,
Differential Topology, 1976. ,
DOI : 10.1007/978-1-4684-9449-5
Principal component analysis, 2002. ,
Generating well-shaped d-dimensional Delaunay Meshes, Theoretical Computer Science, vol.296, issue.1, pp.145-165, 2003. ,
DOI : 10.1016/S0304-3975(02)00437-1
Anatomy of protein pockets and cavities: Measurement of binding site geometry and implications for ligand design, Protein Science, vol.245, issue.9, pp.1884-1897, 1998. ,
DOI : 10.1042/bj3300533
Any open bounded subset of has the same homotopy type as its medial axis, Computer-Aided Design, vol.36, issue.11, pp.1029-1046, 2004. ,
DOI : 10.1016/j.cad.2004.01.011
Dimensionality reduction via subspace and submanifold learning, IEEE Signal Processing Magazine, vol.28, issue.2, 2011. ,
DOI : 10.1109/msp.2010.940005
URL : http://ieeexplore.ieee.org:80/stamp/stamp.jsp?tp=&arnumber=5714387
The Gudhi Library: Simplicial Complexes and Persistent Homology, The 4th Int. Congress on Math. Software, 2014. ,
DOI : 10.1007/978-3-662-44199-2_28
URL : https://hal.archives-ouvertes.fr/hal-01108461
Topology representing networks, Neural Networks, vol.7, issue.3, pp.507-522, 1994. ,
DOI : 10.1016/0893-6080(94)90109-0
Morse Theory, 2006. ,
A constructive proof of the generalized Lovász lemma, J. ACM, vol.57, issue.2, p.2010 ,
Randomized Algorithms, 1995. ,
Computational Geometry: An Introduction Through Randomized Algorithms, 1994. ,
Elementary differential topology, 1966. ,
Elements of algebraic topology, 1984. ,
Finding the Homology of Submanifolds with High Confidence from??Random??Samples, Discrete & Computational Geometry, vol.33, issue.11, pp.419-441, 2008. ,
DOI : 10.1007/b97315
Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 1992. ,
Persistence Theory: From Quiver Representations to Data Analysis, AMS Mathematical Surveys and Monographs, vol.209, 2015. ,
DOI : 10.1090/surv/209
URL : https://hal.archives-ouvertes.fr/hal-01247501
Geometry, a comprehensive course, 1970. ,
A unified approach to the change of resolution: space and gray-level, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.11, issue.7, pp.739-742, 1989. ,
DOI : 10.1109/34.192468
Semiconcave Functions in Alexandrov???s Geometry, Surveys in Differential Geometry: Metric and Comparison Geometry, 2007. ,
DOI : 10.4310/SDG.2006.v11.n1.a6
URL : http://www.intlpress.com/site/pub/files/_fulltext/journals/sdg/2006/0011/0001/SDG-2006-0011-0001-a006.pdf
Towards computing homology from finite approximations, Topology Proceedings, pp.503-532, 1999. ,
Convex Analysis, 1970. ,
DOI : 10.1515/9781400873173
The Earth Mover's Distance as a Metric for Image Retrieval, International Journal of Computer Vision, vol.40, issue.2, pp.99-121, 2000. ,
DOI : 10.1023/A:1026543900054
The upper bound theorem for polytopes: an easy proof of its asymptotic version, Computational Geometry, vol.5, issue.2, pp.115-116, 1995. ,
DOI : 10.1016/0925-7721(95)00013-Y
Star splaying, Proceedings of the twenty-first annual symposium on Computational geometry , SCG '05, pp.237-246, 2005. ,
DOI : 10.1145/1064092.1064129
A vector identity for the Dirichlet tessellation, Mathematical Proceedings of the Cambridge Philosophical Society, vol.21, issue.01, pp.151-155, 1980. ,
DOI : 10.2307/1425985
A brief description of natural neighbour interpolation, Interpreting Multivariate Data, pp.21-36, 1981. ,
Robin Moser makes Lovász Local Lemma Algorithmic! https, 2009. ,
Numerical linear algebra, Society for Industrial and Applied Mathematics, 1997. ,
DOI : 10.1137/1.9780898719574
Topics in Optimal Transportation, 2003. ,
DOI : 10.1090/gsm/058
On C 1 -Complexes, The Annals of Mathematics, vol.41, issue.4, pp.809-824, 1940. ,
DOI : 10.2307/1968861
Geometric integration theory, 1957. ,
DOI : 10.1515/9781400877577
Lectures on Polytopes, Graduate Texts in Mathematics, vol.152, 1994. ,
DOI : 10.1007/978-1-4613-8431-1