{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T12:29:51Z","timestamp":1764937791460,"version":"3.41.0"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,8,28]],"date-time":"2024-08-28T00:00:00Z","timestamp":1724803200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,8,28]],"date-time":"2024-08-28T00:00:00Z","timestamp":1724803200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000104","name":"National Aeronautics and Space Administration","doi-asserted-by":"publisher","award":["80NSSC20M0163"],"award-info":[{"award-number":["80NSSC20M0163"]}],"id":[{"id":"10.13039\/100000104","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CBET-2112085"],"award-info":[{"award-number":["CBET-2112085"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2025,7]]},"DOI":"10.1007\/s00454-024-00683-5","type":"journal-article","created":{"date-parts":[[2024,8,28]],"date-time":"2024-08-28T14:04:59Z","timestamp":1724853899000},"page":"203-241","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications"],"prefix":"10.1007","volume":"74","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8209-3604","authenticated-orcid":false,"given":"Thomas","family":"Lew","sequence":"first","affiliation":[]},{"given":"Riccardo","family":"Bonalli","sequence":"additional","affiliation":[]},{"given":"Lucas","family":"Janson","sequence":"additional","affiliation":[]},{"given":"Marco","family":"Pavone","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,8,28]]},"reference":[{"key":"683_CR1","unstructured":"Aamari, E.: Convergence Rates for Geometric Inference. PhD Thesis, Universit\u00e9 Paris-Saclay (2017). https:\/\/theses.hal.science\/tel-01607782"},{"key":"683_CR2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10208-022-09588-2","volume":"24","author":"E Aamari","year":"2022","unstructured":"Aamari, E., Knop, A.: Adversarial manifold estimation. Found. Comput. Math. 24, 1\u201397 (2022)","journal-title":"Found. Comput. Math."},{"issue":"1","key":"683_CR3","doi-asserted-by":"crossref","first-page":"1359","DOI":"10.1214\/19-EJS1551","volume":"13","author":"E Aamari","year":"2019","unstructured":"Aamari, E., Kim, J., Chazal, F., Michel, B., Rinaldo, A., Wasserman, L.: Estimating the reach of a manifold. Electron. J. Stat. 13(1), 1359\u20131399 (2019)","journal-title":"Electron. J. Stat."},{"issue":"4","key":"683_CR4","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1007\/s00454-017-9962-z","volume":"59","author":"E Aamari","year":"2018","unstructured":"Aamari, E., Levrard, C.: Stability and minimax optimality of tangential Delaunay complexes for manifold reconstruction. Discrete Comput. Geom. 59(4), 923\u2013971 (2018)","journal-title":"Discrete Comput. Geom."},{"issue":"527","key":"683_CR5","doi-asserted-by":"crossref","first-page":"1162","DOI":"10.1080\/01621459.2018.1482751","volume":"114","author":"E Arias-Castro","year":"2019","unstructured":"Arias-Castro, E., Pateiro-Lopez, B., Rodriguez-Casal, A.: Minimax estimation of the volume of a set under the rolling ball condition. J. Am. Stat. Assoc. 114(527), 1162\u20131173 (2019)","journal-title":"J. Am. Stat. Assoc."},{"issue":"4","key":"683_CR6","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1239\/aap\/1011994024","volume":"33","author":"A Baillo","year":"2001","unstructured":"Baillo, A., Cuevas, A.: On the estimation of a star-shaped set. Adv. Appl. Probab. 33(4), 717\u2013726 (2001)","journal-title":"Adv. Appl. Probab."},{"issue":"4","key":"683_CR7","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1287\/moor.23.4.769","volume":"23","author":"A Ben-Tal","year":"1998","unstructured":"Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23(4), 769\u2013805 (1998)","journal-title":"Math. Oper. Res."},{"issue":"2","key":"683_CR8","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/s00454-021-00290-8","volume":"67","author":"C Berenfeld","year":"2021","unstructured":"Berenfeld, C., Harvey, J., Hoffmann, M., Shankar, K.: Estimating the reach of a manifold via its convexity defect function. Discrete Comput. Geom. 67(2), 403\u2013438 (2021)","journal-title":"Discrete Comput. Geom."},{"issue":"3","key":"683_CR9","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1137\/080734510","volume":"53","author":"D Bertsimas","year":"2011","unstructured":"Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464\u2013501 (2011)","journal-title":"SIAM Rev."},{"key":"683_CR10","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1007\/s10208-017-9344-1","volume":"18","author":"J-D Boissonnat","year":"2018","unstructured":"Boissonnat, J.-D., Dyer, R., Ghosh, A.: Delaunay triangulation of manifolds. Found. Comput. Math. 18, 399\u2013431 (2018)","journal-title":"Found. Comput. Math."},{"issue":"1","key":"683_CR11","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s00454-013-9557-2","volume":"51","author":"J-D Boissonnat","year":"2013","unstructured":"Boissonnat, J.-D., Ghosh, A.: Manifold reconstruction using tangential Delaunay complexes. Discrete Comput. Geom. 51(1), 221\u2013267 (2013)","journal-title":"Discrete Comput. Geom."},{"issue":"1\u20132","key":"683_CR12","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s41468-019-00029-8","volume":"3","author":"J-D Boissonnat","year":"2019","unstructured":"Boissonnat, J.-D., Lieutier, A., Wintraecken, M.: The reach, metric distortion, geodesic convexity and the variation of tangent spaces. J. Appl. Comput. Topol. 3(1\u20132), 29\u201358 (2019)","journal-title":"J. Appl. Comput. Topol."},{"issue":"2","key":"683_CR13","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1239\/aap\/1035228070","volume":"30","author":"H Br\u00e4ker","year":"1998","unstructured":"Br\u00e4ker, H., Hsing, T., Bingham, N.H.: On the Hausdorff distance between a convex set and an interior random convex hull. Adv. Appl. Probab. 30(2), 295\u2013316 (1998)","journal-title":"Adv. Appl. Probab."},{"issue":"110","key":"683_CR14","first-page":"3603","volume":"16","author":"F Chazal","year":"2015","unstructured":"Chazal, F., Glisse, M., Labru\u00e8re, C., Michel, B.: Convergence rates for persistence diagram estimation in topological data analysis. J. Mach. Learn. Res. 16(110), 3603\u20133635 (2015)","journal-title":"J. Mach. Learn. Res."},{"issue":"5","key":"683_CR15","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1111\/rssb.12149","volume":"78","author":"A Cholaquidis","year":"2016","unstructured":"Cholaquidis, A., Fraiman, R., Lugosi, G., Pateiro-L\u00f3pez, B.: Set estimation from reflected Brownian motion. J. R. Stat. Soc. B 78(5), 1057\u20131078 (2016)","journal-title":"J. R. Stat. Soc. B"},{"key":"683_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-023-00624-8","author":"R Cotsakis","year":"2024","unstructured":"Cotsakis, R.: Computable bounds for the reach and r-convexity of subsets of $${{\\mathbb{R} }}^d$$. Discrete Comput. Geom. (2024). https:\/\/doi.org\/10.1007\/s00454-023-00624-8","journal-title":"Discrete Comput. Geom."},{"issue":"2","key":"683_CR17","first-page":"71","volume":"25","author":"A Cuevas","year":"2009","unstructured":"Cuevas, A.: Set estimation: another bridge between statistics and geometry. Bol. Estadistica Investig. Oper. 25(2), 71\u201385 (2009)","journal-title":"Bol. Estadistica Investig. Oper."},{"issue":"3","key":"683_CR18","doi-asserted-by":"crossref","first-page":"700","DOI":"10.2307\/3215149","volume":"31","author":"L De Haan","year":"1994","unstructured":"De Haan, L., Resnick, S.: Estimating the home range. Appl. Probab. 31(3), 700\u2013720 (1994)","journal-title":"Appl. Probab."},{"issue":"2","key":"683_CR19","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.acha.2013.11.003","volume":"37","author":"E De Vito","year":"2014","unstructured":"De Vito, E., Rosasco, L., Toigo, A.: Learning sets with separating kernels. Appl. Comput. Harmon. Anal. 37(2), 185\u2013217 (2014)","journal-title":"Appl. Comput. Harmon. Anal."},{"issue":"3","key":"683_CR20","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1137\/0138038","volume":"38","author":"L Devroye","year":"1980","unstructured":"Devroye, L., Wise, G.L.: Detection of abnormal behavior via nonparametric estimation of the support. SIAM J. Appl. Math. 38(3), 480\u2013488 (1980)","journal-title":"SIAM J. Appl. Math."},{"issue":"2","key":"683_CR21","doi-asserted-by":"crossref","first-page":"384","DOI":"10.2307\/1428063","volume":"28","author":"L D\u00fcmbgen","year":"1996","unstructured":"D\u00fcmbgen, L., Walther, G.: Rates of convergence for random approximations of convex sets. Adv. Appl. Probab. 28(2), 384\u2013393 (1996)","journal-title":"Adv. Appl. Probab."},{"key":"683_CR22","doi-asserted-by":"crossref","first-page":"163938","DOI":"10.1109\/ACCESS.2021.3133370","volume":"9","author":"M Everett","year":"2021","unstructured":"Everett, M., Habibi, G., Chuangchuang, S., How, J.P.: Reachability analysis of neural feedback loops. IEEE Access 9, 163938\u2013163953 (2021)","journal-title":"IEEE Access"},{"key":"683_CR23","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1090\/S0002-9947-1959-0110078-1","volume":"93","author":"H Federer","year":"1959","unstructured":"Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418\u2013491 (1959)","journal-title":"Trans. Am. Math. Soc."},{"issue":"3","key":"683_CR24","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1080\/00029890.1990.11995583","volume":"97","author":"GB Folland","year":"1990","unstructured":"Folland, G.B.: Remainder estimates in Taylor\u2019s theorem. Am. Math. Mon. 97(3), 233\u2013235 (1990)","journal-title":"Am. Math. Mon."},{"issue":"1","key":"683_CR25","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s11004-009-9257-x","volume":"42","author":"A Gonz\u00e1lez","year":"2009","unstructured":"Gonz\u00e1lez, A.: Measurement of areas on a sphere using Fibonacci and latitude\u2013longitude lattices. Math. Geosci. 42(1), 49\u201364 (2009)","journal-title":"Math. Geosci."},{"issue":"3","key":"683_CR26","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/s11222-007-9027-x","volume":"17","author":"W Jang","year":"2007","unstructured":"Jang, W., Hendry, M.: Cluster analysis of massive datasets in astronomy. Stat. Comput. 17(3), 253\u2013262 (2007)","journal-title":"Stat. Comput."},{"key":"683_CR27","volume-title":"Introduction to Smooth Manifolds","author":"JM Lee","year":"2012","unstructured":"Lee, J.M.: Introduction to Smooth Manifolds, 2nd edn. Springer, New York (2012)","edition":"2"},{"key":"683_CR28","volume-title":"Introduction to Riemannian Manifolds","author":"JM Lee","year":"2018","unstructured":"Lee, J.M.: Introduction to Riemannian Manifolds, 2nd edn. Springer, Cham (2018)","edition":"2"},{"key":"683_CR29","unstructured":"Lew, T., Janson, L., Bonalli, R., Pavone, M.: A simple and efficient sampling-based algorithm for general reachability analysis. In: Learning for Dynamics and Control Conference (2022)"},{"key":"683_CR30","unstructured":"Lew, T., Pavone, M.: Sampling-based reachability analysis: a random set theory approach with adversarial sampling. In: Conference on Robot Learning (2020)"},{"issue":"5","key":"683_CR31","doi-asserted-by":"crossref","first-page":"2888","DOI":"10.1109\/TRO.2022.3154715","volume":"38","author":"T Lew","year":"2022","unstructured":"Lew, T., Sharma, A., Harrison, J., Bylard, A., Pavone, M.: Safe active dynamics learning and control: a sequential exploration\u2013exploitation framework. IEEE Trans. Robot. 38(5), 2888\u20132907 (2022)","journal-title":"IEEE Trans. Robot."},{"issue":"2","key":"683_CR32","first-page":"342","volume":"58","author":"S Leyffer","year":"2020","unstructured":"Leyffer, S., Menickelly, M., Munson, T., Vanaret, C., Wild, S.M.: A survey of nonlinear robust optimization. INFOR Inf. Syst. Oper. Res. 58(2), 342\u2013373 (2020)","journal-title":"INFOR Inf. Syst. Oper. Res."},{"key":"683_CR33","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4471-7349-6","volume-title":"Theory of Random Sets","author":"I Molchanov","year":"2017","unstructured":"Molchanov, I.: Theory of Random Sets, 2nd edn. Springer, London (2017)","edition":"2"},{"issue":"1","key":"683_CR34","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1007\/s00454-008-9053-2","volume":"39","author":"P Niyogi","year":"2008","unstructured":"Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39(1), 419\u2013441 (2008)","journal-title":"Discrete Comput. Geom."},{"key":"683_CR35","unstructured":"Pateiro-L\u00f3pez, B.: Set estimation under convexity type restrictions. PhD Thesis, Universidade de Santiago de Compostela (2008). http:\/\/eio.usc.es\/pub\/pateiro\/files\/THESIS_BeatrizPateiroLopez.pdf"},{"issue":"4","key":"683_CR36","doi-asserted-by":"crossref","first-page":"501","DOI":"10.4310\/jdg\/1214432545","volume":"9","author":"J Rauch","year":"1974","unstructured":"Rauch, J.: An inclusion theorem for ovaloids with comparable second fundamental forms. J. Differ. Geom. 9(4), 501\u2013505 (1974)","journal-title":"J. Differ. Geom."},{"issue":"3","key":"683_CR37","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.cviu.2003.10.002","volume":"93","author":"A Ray Chaudhuri","year":"2004","unstructured":"Ray Chaudhuri, A., Basu, A., Tan, K., Bhandari, S., Chaudhuri, B.B.: An efficient set estimator in high dimensions: consistency and applications to fast data visualization. Comput. Vis. Image Underst. 93(3), 260\u2013287 (2004)","journal-title":"Comput. Vis. Image Underst."},{"key":"683_CR38","doi-asserted-by":"crossref","first-page":"483","DOI":"10.2307\/3213451","volume":"14","author":"BD Ripley","year":"1977","unstructured":"Ripley, B.D., Rasson, J.P.: Finding the edge of a Poisson forest. J. Appl. Probab. 14, 483\u2013491 (1977)","journal-title":"J. Appl. Probab."},{"issue":"1","key":"683_CR39","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1051\/ps\/2016015","volume":"20","author":"A Rodriguez-Casal","year":"2016","unstructured":"Rodriguez-Casal, A., Saavedra-Nieves, P.: A fully data-driven method for estimating the shape of a point cloud. ESAIM Probab. Stat. 20(1), 332\u2013348 (2016)","journal-title":"ESAIM Probab. Stat."},{"key":"683_CR40","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3389\/fams.2017.00023","volume":"3","author":"A Rudi","year":"2017","unstructured":"Rudi, A., De Vito, E., Verri, A., Odone, F.: Regularized kernel algorithms for support estimation. Front. Appl. Math. Stat. 3, 1\u201315 (2017)","journal-title":"Front. Appl. Math. Stat."},{"issue":"1","key":"683_CR41","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1007\/BF02311318","volume":"32","author":"R Schneider","year":"1987","unstructured":"Schneider, R.: Approximation of convex bodies by random polytopes. Aequationes Math. 32(1), 304\u2013310 (1987)","journal-title":"Aequationes Math."},{"issue":"3","key":"683_CR42","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1111\/j.1365-2818.1988.tb04682.x","volume":"151","author":"R Schneider","year":"1988","unstructured":"Schneider, R.: Random approximation of convex sets. J. Microsc. 151(3), 211\u2013227 (1988)","journal-title":"J. Microsc."},{"key":"683_CR43","volume-title":"Convex Bodies: The Brunn\u2013Minkowski Theory","author":"R Schneider","year":"2014","unstructured":"Schneider, R.: Convex Bodies: The Brunn\u2013Minkowski Theory, 2nd edn. Cambridge University Press, Cambridge (2014)","edition":"2"},{"key":"683_CR44","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-78859-1","volume-title":"Stochastic and Integral Geometry","author":"R Schneider","year":"2008","unstructured":"Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2008)"},{"issue":"4","key":"683_CR45","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1002\/(SICI)1099-1476(19990310)22:4<301::AID-MMA42>3.0.CO;2-M","volume":"22","author":"R Schneider","year":"1999","unstructured":"Schneider, R.: On a generalization of Blaschke\u2019s rolling theorem and the smoothing of surfaces. Math. Methods Appl. Sci. 22(4), 301\u2013316 (1999)","journal-title":"Math. Methods Appl. Sci."},{"key":"683_CR46","doi-asserted-by":"crossref","unstructured":"Sch\u00fcrmann, B., Kochdumper, N., Althoff, M.: Reachset model predictive control for disturbed nonlinear systems. In: Proceedings of the IEEE Conference on Decision and Control (2018)","DOI":"10.1109\/CDC.2018.8619781"},{"key":"683_CR47","doi-asserted-by":"crossref","DOI":"10.1016\/j.ejcon.2022.100680","volume":"68","author":"J Sieber","year":"2022","unstructured":"Sieber, J., Zanelli, A., Bennani, S., Zeilinger, M.N.: System level disturbance reachable sets and their application to tube-based MPC. Eur. J. Control 68, 100680 (2022)","journal-title":"Eur. J. Control"},{"issue":"4","key":"683_CR48","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/s12532-020-00179-2","volume":"12","author":"B Stellato","year":"2020","unstructured":"Stellato, B., Banjac, G., Goulart, P., Bemporad, A., Boyd, S.: OSQP: an operator splitting solver for quadratic programs. Math. Program. Comput. 12(4), 637\u2013672 (2020)","journal-title":"Math. Program. Comput."},{"issue":"6","key":"683_CR49","doi-asserted-by":"crossref","first-page":"2273","DOI":"10.1214\/aos\/1030741072","volume":"25","author":"G Walther","year":"1997","unstructured":"Walther, G.: Granulometric smoothing. Ann. Stat. 25(6), 2273\u20132299 (1997)","journal-title":"Ann. Stat."},{"issue":"12","key":"683_CR50","doi-asserted-by":"crossref","first-page":"e0243330","DOI":"10.1371\/journal.pone.0243330","volume":"15","author":"PM Wensing","year":"2020","unstructured":"Wensing, P.M., Slotine, J.-J.: Beyond convexity\u2014contraction and global convergence of gradient descent. PLoS ONE 15(12), e0243330 (2020)","journal-title":"PLoS ONE"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-024-00683-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-024-00683-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-024-00683-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T21:14:19Z","timestamp":1750281259000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-024-00683-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,28]]},"references-count":50,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["683"],"URL":"https:\/\/doi.org\/10.1007\/s00454-024-00683-5","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"type":"print","value":"0179-5376"},{"type":"electronic","value":"1432-0444"}],"subject":[],"published":{"date-parts":[[2024,8,28]]},"assertion":[{"value":"10 July 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 February 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 July 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 August 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}