N. Alon and J. H. Spencer, 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., [51, Theorem 11, 2008.

N. Amenta and M. Bern, Surface Reconstruction by Voronoi Filtering, Discrete & Computational Geometry, vol.22, issue.4, pp.481-504, 1999.
DOI : 10.1007/PL00009475

D. Attali, J. Boissonnat, and H. Edelsbrunner, Stability and Computation of Medial Axes - a State-of-the-Art Report, Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration of Mathematics and Vizualization, pp.109-125, 2009.
DOI : 10.1007/b106657_6

URL : https://hal.archives-ouvertes.fr/hal-00468690

D. Attali, H. Edelsbrunner, and Y. Mileyko, 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

D. Attali, A. Lieutier, and D. Salinas, 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

F. Aurenhammer, Power Diagrams: Properties, Algorithms and Applications, SIAM Journal on Computing, vol.16, issue.1, pp.78-96, 1987.
DOI : 10.1137/0216006

F. Aurenhammer, R. Klein, and D. Lee, Voronoi Diagrams and Delaunay Triangulations, World Scientific, 2013.
DOI : 10.1142/8685

C. Paul, B. , and K. Rao, A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields, Journal of the ACM, vol.42, issue.1, pp.67-90, 1995.

M. Berger, Géométrie (vols. 1-5) Fernand Nathan, p.295, 1977.

G. Biau, F. Chazal, D. Cohen-steiner, L. Devroye, and C. Rodriguez, 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

. Andrewj, I. Blumberg, M. Gal, M. Mandell, and . Pancia, Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces, Foundations of Computational Mathematics, vol.14, issue.4, pp.745-789, 2014.

J. Boissonnat, C. S. Karthik, and S. Tavenas, Building Efficient and Compact Data Structures for Simplicial Complexes, International Symposium on Computational Geometry 2015, 2015.
DOI : 10.1145/253168.253192

URL : https://hal.archives-ouvertes.fr/hal-01145407

J. Boissonnat, O. Devillers, K. Dutta, and M. Glisse, Delaunay triangulation of a random sample of a good sample has linear size, 2017.

J. Boissonnat, O. Devillers, S. Pion, M. Teillaud, and M. Yvinec, 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/inria-00167199

J. Boissonnat, R. Dyer, and A. Ghosh, Delaunay Triangulation of Manifolds, Foundations of Computational Mathematics, vol.45, issue.2, 2013.
DOI : 10.1145/336154.336221

URL : https://hal.archives-ouvertes.fr/hal-01509888

J. Boissonnat, R. Dyer, and A. Ghosh, 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

J. Boissonnat, R. Dyer, and A. Ghosh, 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-01022371

J. Boissonnat, R. Dyer, and A. Ghosh, A Probabilistic Approach to Reducing Algebraic Complexity of Delaunay Triangulations, Algorithms -ESA 2015 -23rd Annual European Symposium Proceedings, pp.595-606, 2015.
DOI : 10.1007/s10711-008-9261-1

URL : https://hal.archives-ouvertes.fr/hal-01213070

J. Boissonnat and R. Dyer, Arijit Ghosh, and Mathijs Wintraecken. Local criteria for triangulation of manifolds, 2017.

J. Boissonnat and J. Flötotto, 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

J. Boissonnat and A. Ghosh, Manifold reconstruction using tangential delaunay complexes, Proc. 26th Annual Symposium on Computational Geometry, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00932209

J. Boissonnat and A. Ghosh, Manifold reconstruction using tangential Delaunay complexes, Discrete and computational Geometry, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00932209

J. Boissonnat and C. Maria, 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

J. Boissonnat, F. Nielsen, and R. Nock, Bregman Voronoi Diagrams, Discrete & Computational Geometry, vol.12, issue.2, p.2010
DOI : 10.1515/9781400873173

URL : https://hal.archives-ouvertes.fr/hal-00488441

J. Boissonnat and S. Oudot, 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

J. Boissonnat, C. Wormser, and M. Yvinec, 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

J. Boissonnat, C. Wormser, and M. Yvinec, Locally uniform anisotropic meshing, Proceedings of the twenty-fourth annual symposium on Computational geometry , SCG '08, pp.270-277, 2008.
DOI : 10.1145/1377676.1377724

URL : https://hal.archives-ouvertes.fr/inria-00275430

J. Boissonnat and M. Yvinec, Algorithmic Geometry, 1998.
DOI : 10.1017/CBO9781139172998

F. Bolley, A. Guillin, and C. Villani, 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

M. Buchet, F. Chazal, Y. Steve, . Oudot, R. Donald et al., Efficient and robust persistent homology for measures, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.168-180, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01074566

D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry
DOI : 10.1090/gsm/033

C. Caillerie, F. Chazal, J. Dedecker, and B. Michel, 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

S. S. Cairns, A simple triangulation method for smooth maniolds, Bull. Amer. Math. Soc, vol.67, pp.380-390, 1961.

P. Cannarsa and C. Cinestrari, Semiconcave Functions, Hamilton- Jacobi Equations, and Optimal Control, Brikhauser, vol.58, 2004.

R. Chaine, A geometric convection approach of 3d-reconstruction, 1st Symposium on Geometry Processing, pp.218-229, 2003.
URL : https://hal.archives-ouvertes.fr/inria-00071898

F. Chazal, D. Cohen-steiner, L. J. Guibas, M. Glisse, and S. Y. Oudot, Proximity of persistence modules and their diagrams, Proceedings of the 25th annual symposium on Computational geometry, SCG '09, 2009.
DOI : 10.1145/1542362.1542407

URL : https://hal.archives-ouvertes.fr/inria-00292566

F. Chazal, D. Cohen-steiner, L. J. Guibas, F. Memoli, and S. Y. Oudot, 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

F. Chazal, D. Cohen-steiner, and A. Lieutier, 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

F. Chazal, D. Cohen-steiner, A. Lieutier, and B. Thibert, Shape smoothing using double offsets, Proceedings of the 2007 ACM symposium on Solid and physical modeling , SPM '07, 2007.
DOI : 10.1145/1236246.1236273

URL : https://hal.archives-ouvertes.fr/hal-00171507

F. Chazal, D. Cohen-steiner, A. Lieutier, and B. Thibert, 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/hal-00864486

F. Chazal, D. Cohen-steiner, and Q. Mérigot, Geometric Inference for Probability Measures, Foundations of Computational Mathematics, vol.40, issue.2, pp.733-751, 2011.
DOI : 10.1090/gsm/058

URL : https://hal.archives-ouvertes.fr/hal-00772444

F. Chazal, V. De-silva, M. Glisse, and S. Oudot, The structure and stability of persistence modules, p.2012
DOI : 10.1007/978-3-319-42545-0

URL : https://hal.archives-ouvertes.fr/hal-01330678

F. Chazal, V. De-silva, and S. Oudot, 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

F. Chazal and A. Lieutier, The ?????-medial axis???, Graphical Models, vol.67, issue.4, pp.304-331, 2005.
DOI : 10.1016/j.gmod.2005.01.002

F. Chazal and A. Lieutier, Weak feature size and persistent homology, Proceedings of the twenty-first annual symposium on Computational geometry , SCG '05
DOI : 10.1145/1064092.1064132

F. Chazal and A. Lieutier, 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

F. Chazal and S. Y. Oudot, 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

F. Chazal, D. Cohen-steiner, and A. Lieutier, 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

F. Chazal, B. Fasy, F. Lecci, B. Michel, A. Rinaldo et al., Subsampling methods for persistent homology, Proceedings of the 32nd International Conference on Machine Learning (ICML-15) Conference Proceedings, pp.2143-2151, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01073073

F. Chazal, T. Brittany, F. Fasy, B. Lecci, A. Michel et al., Robust topological inference: Distance to a measure and kernel distance. arXiv preprint, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01232217

F. Chazal, M. Glisse, C. , and B. Michel, Convergence rates for persistence diagram estimation in topological data analysis, Journal of Machine Learning Research, vol.16, pp.3603-3635, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01284275

F. Chazal, P. Massart, and B. Michel, Rates of convergence for robust geometric inference. arXiv preprint, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01336913

B. Chazelle, An optimal convex hull algorithm in any fixed dimension, Geometric Topology: recent developments, pp.377-409, 1990.
DOI : 10.1137/1116025

J. Cheeger, W. Müller, and R. Schrader, 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

S. Cheng, T. K. Dey, and E. A. Ramos, Manifold Reconstruction from Point Samples, Proc. ACM-SIAM Symp. Discrete Algorithms, pp.1018-1027, 2005.

S. Cheng, T. K. Dey, and E. A. Ramos, Manifold reconstruction from point samples, Proc. ACM-SIAM Sympos. Discrete Algorithms, pp.1018-1027, 2005.

F. H. Clarke, Optimization and Nonsmooth Analysis, 1983.
DOI : 10.1137/1.9781611971309

K. L. Clarkson and P. W. Shor, Applications of random sampling in computational geometry, II, Discrete & Computational Geometry, vol.1, issue.5, pp.387-421, 1989.
DOI : 10.1145/73393.73407

D. Cohen-steiner, H. Edelsbrunner, and J. Harer, Stability of Persistence Diagrams, Discrete & Computational Geometry, vol.37, issue.1, pp.103-120, 2007.
DOI : 10.1007/s00454-006-1276-5

M. Mark-de-berg, M. Van-kreveld, O. Overmars, and . Schwarzkopf, Computational Geometry: Algorithms and Applications, 2000.

V. De-silva and R. Ghrist, 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

S. Vin-de, A weak characterisation of the delaunay triangulation, Geometriae Dedicata, vol.135, issue.1, pp.39-64, 2008.

B. Delaunay, Sur la sphère vide, Otdelenie Matematicheskii i Estestvennyka Nauk, pp.793-800, 1934.

T. K. Dey, Curve and Surface Reconstruction : Algorithms with Mathematical Analysis, 2007.
DOI : 10.1017/CBO9780511546860

R. Dyer, G. Vegter, and M. Wintraecken, Riemannian simplices and triangulations, 31st International Symposium on Computational Geometry, pp.255-269, 2015.
DOI : 10.1515/9781400877577

H. Edelsbrunner, 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

H. Edelsbrunner, Surface reconstruction by wrapping finite point sets in space, pp.379-404, 2003.

H. Edelsbrunner, M. Facello, P. Fu, and J. Liang, Measuring proteins and voids in proteins, Proceedings of the Twenty-Eighth Hawaii International Conference on System Sciences, vol.5, pp.256-264, 1995.
DOI : 10.1109/HICSS.1995.375331

H. Edelsbrunner, M. Facello, and J. Liang, 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

H. Edelsbrunner, D. Letscher, and A. Zomorodian, Topological Persistence and Simplification, Discrete & Computational Geometry, vol.28, issue.4, pp.511-533, 2002.
DOI : 10.1007/s00454-002-2885-2

H. Edelsbrunner and E. P. Mücke, Three-dimensional alpha shapes, Proceedings of the 1992 workshop on, pp.75-82, 1992.

H. Edelsbrunner, Algorithms in combinatorial geometry, 1987.
DOI : 10.1007/978-3-642-61568-9

H. Edelsbrunner, Geometry and Topology for Mesh Generation, 2001.

H. Edelsbrunner, L. John, and . Harer, Computational topology: an introduction, 2010.
DOI : 10.1090/mbk/069

H. Edelsbrunner, D. Kirkpatrick, and R. Seidel, 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

H. Edelsbrunner, X. Li, G. L. Miller, A. Stathopoulos, D. Talmor et al., 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://citeseerx.ist.psu.edu/viewdoc/summary?doi=

H. Edelsbrunner and H. Wagner, Topological data analysis with bregman divergences. CoRR, abs, 1607.

D. Eppstein, M. Löffler, and D. Strash, Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time, Algorithms and Computation -21st International Symposium, ISAAC 2010 Proceedings, Part I, pp.403-414, 2010.
DOI : 10.1007/s00373-007-0738-8

B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan et al., Confidence sets for persistence diagrams, The Annals of Statistics, vol.42, issue.6, pp.2301-2339, 2014.
DOI : 10.1214/14-AOS1252SUPP

T. Feder and D. Greene, 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

H. Federer, Curvature measures. Transactions of the, pp.418-491, 1959.
DOI : 10.1090/s0002-9947-1959-0110078-1

D. Freedman, 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

P. Frosini and C. Landi, Size theory as a topological tool for computer vision, Pattern Recognition and Image Analysis, vol.9, pp.596-603, 1999.

J. H. Fu, Convergence of curvatures in secant approximations, Journal of Differential Geometry, vol.37, issue.1, pp.177-190, 1993.
DOI : 10.4310/jdg/1214453427

J. H. Fu, Tubular neighborhoods in Euclidean spaces, Duke Mathematical Journal, vol.52, issue.4, pp.1025-1046, 1985.
DOI : 10.1215/S0012-7094-85-05254-8

W. Fulton, Algebraic Topology: a First Course, 1995.
DOI : 10.1007/978-1-4612-4180-5

J. Gallier, Notes on Convex Sets, Polytopes, Polyhedra Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations, 2007.
URL : https://hal.archives-ouvertes.fr/inria-00193831

S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, 1990.
URL : https://hal.archives-ouvertes.fr/hal-00002870

J. Giesen and M. John, The flow complex: A data structure for geometric modeling, Proc. 14th ACM-SIAM Sympos. Discrete Algorithms (SODA), pp.285-294, 2003.
DOI : 10.1016/j.comgeo.2007.01.002

J. Giesen and U. Wagner, 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

T. Gonzales, 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

K. Grove, Critical point theory for distance functions, Proc. of Symposia in Pure Mathematics, 1993.
DOI : 10.1090/pspum/054.3/1216630

L. Guibas, D. Morozov, and Q. Mérigot, 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-00569029

S. Har-peled, Geometric approximation algorithms, 2011.
DOI : 10.1090/surv/173

S. Har-peled and M. Mendel, 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

A. Hatcher, Algebraic Topology, 2002.

M. W. Hirsch, Differential Topology, 1976.
DOI : 10.1007/978-1-4684-9449-5

I. T. Jolliffe, Principal component analysis, 2002.

X. Li, 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

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

X. Li, 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

J. Liang, H. Edelsbrunner, and C. Woodward, Anatomy of protein pockets and cavities: measurement of binding site geometry and implications for ligand design Any open bounded subset of R n has the same homotopy type as it medial axis, Protein Science Computer-Aided Design, vol.7, issue.11, pp.1884-1897, 1998.

A. Lieutier, 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

Y. Ma, P. Niyogi, G. Sapiro, and R. Vidal-ed, 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

C. Maria, J. Boissonnat, M. Glisse, and M. Yvinec, The Gudhi Library: Simplicial Complexes and Persistent Homology, The 4th International Congress on Mathematical Software (ICMS), 2014.
DOI : 10.1007/978-3-662-44199-2_28

URL : https://hal.archives-ouvertes.fr/hal-01108461

T. Martinetz and K. Schulten, Topology representing networks, Neural Networks, vol.7, issue.3, pp.507-522, 1994.
DOI : 10.1016/0893-6080(94)90109-0

R. A. Moser and G. Tardos, A constructive proof of the generalized Lovász lemma, Journal of the ACM, issue.2, p.2010

R. Motwani and P. Raghavan, Randomized Algorithms, 1995.

K. Mulmuley, Computational Geometry: An Introduction Through Randomized Algorithms, 1994.

J. R. Munkres, Elementary differential topology, 1966.

J. R. Munkres, Elements of algebraic topology, 1984.

P. Niyogi, S. Smale, and S. Weinberger, 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

A. Okabe, B. Boots, and K. Sugihara, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 1992.

S. Y. Oudot, 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

D. Pedoe, Geometry, a comprehensive course, 1970.

S. Peleg, M. Werman, and H. Rom, 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

A. Petrunin, Semiconcave Functions in Alexandrov???s Geometry, Surveys in Differential Geometry: Metric and Comparison Geometry, 2007.
DOI : 10.4310/SDG.2006.v11.n1.a6

V. Robins, Toxard computing homology from finite approximations, Topology Proceedings, pp.503-532, 1999.

R. T. Rockafellar, Convex Analysis, 1970.
DOI : 10.1515/9781400873173

Y. Rubner, C. Tomasi, and L. J. Guibas, 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

R. Seidel, 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

R. Seidel, A convex hull algorithm optimal for point sets in even dimensions, 1981.

J. Shewchuk, Star splaying, Proceedings of the twenty-first annual symposium on Computational geometry , SCG '05, pp.237-246, 2005.
DOI : 10.1145/1064092.1064129

R. Sibson, 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

]. R. Sibson, A brief description of natural neighbour interpolation, Interpreting Multivariate Data, pp.21-36, 1981.

J. Spencer, Robin Moser makes Lovász Local Lemma Algorithmic

L. N. Trefethen and D. Bau, Numerical linear algebra, Society for Industrial Mathematics, 1997.
DOI : 10.1137/1.9780898719574

C. Villani, Topics in Optimal Transportation, 2003.
DOI : 10.1090/gsm/058

E. Welzl, Smallest enclosing disks (balls and ellipsoids), New Results and New Trends in Computer Science Lecture Notes Comput. Sci, vol.555, pp.359-370, 1991.
DOI : 10.1007/BFb0038202

J. H. Whitehead, On C 1 -Complexes, The Annals of Mathematics, vol.41, issue.4, pp.809-824, 1940.
DOI : 10.2307/1968861

H. Whitney, Geometric integration theory, 1957.
DOI : 10.1515/9781400877577

M. Wintraecken, Bounds on the angle between tangent spaces and the metric distortion for C 2 manifolds with given positive reach, European Workshop on Computational Geometry, p.2017

G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, vol.152, 1994.
DOI : 10.1007/978-1-4613-8431-1