{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T23:40:01Z","timestamp":1749512401711,"version":"3.41.0"},"reference-count":70,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,4,23]],"date-time":"2025-04-23T00:00:00Z","timestamp":1745366400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,23]],"date-time":"2025-04-23T00:00:00Z","timestamp":1745366400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"International Center of Research and Postgraduate Training in Mathematics","award":["ICRTM04_2024.03"],"award-info":[{"award-number":["ICRTM04_2024.03"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2025,7]]},"DOI":"10.1007\/s10589-025-00682-z","type":"journal-article","created":{"date-parts":[[2025,4,23]],"date-time":"2025-04-23T20:13:50Z","timestamp":1745439230000},"page":"1373-1413","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Inner $$\\delta $$-approximation of the convex hull of finite sets"],"prefix":"10.1007","volume":"91","author":[{"given":"Nam-D\u0169ng","family":"Hoang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nguyen Kieu","family":"Linh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4835-1275","authenticated-orcid":false,"given":"Hoang Xuan","family":"Phu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,23]]},"reference":[{"key":"682_CR1","first-page":"96","volume":"119","author":"N Agarwal","year":"2020","unstructured":"Agarwal, N., Brukhim, N., Hazan, E., Lu, Z.: Boosting for control of dynamical systems. Proc. Mach. Learn. Res. 119, 96\u2013103 (2020)","journal-title":"Proc. Mach. Learn. Res."},{"key":"682_CR2","first-page":"4140","volume":"139","author":"E Hazan","year":"2021","unstructured":"Hazan, E., Singh, K.: Boosting for online convex optimization. Proc. Mach. Learn. Res. 139, 4140\u20134149 (2021)","journal-title":"Proc. Mach. Learn. Res."},{"key":"682_CR3","doi-asserted-by":"crossref","unstructured":"Aziz, M. Z., Mertsching, B., Munir, A.: Classification using scale and rotation tolerant shape signatures from convex hulls, Pattern Recognition and Image Analysis, Part II, pp. 663\u2013672, Lecture Notes in Comput. Sci., 3687, Springer, Berlin (2005)","DOI":"10.1007\/11552499_73"},{"key":"682_CR4","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1016\/j.patrec.2019.10.007","volume":"128","author":"H Cevikalp","year":"2019","unstructured":"Cevikalp, H.: High-dimensional data clustering by using local affine\/convex hulls. Pattern Recognit. Lett. 128, 427\u2013432 (2019)","journal-title":"Pattern Recognit. Lett."},{"key":"682_CR5","doi-asserted-by":"publisher","first-page":"792","DOI":"10.1109\/TNNLS.2017.2648038","volume":"29","author":"S Ding","year":"2018","unstructured":"Ding, S., Nie, X., Qiao, H., Zhang, B.: A fast algorithm of convex hull vertices selection for online classification. IEEE Trans. Neural Netw. Learn. Syst. 29, 792\u2013806 (2018)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"682_CR6","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1134\/S1054661819030167","volume":"29","author":"AP Nemirko","year":"2019","unstructured":"Nemirko, A.P.: Lightweight nearest convex hull classifier. Pattern Recognit Image Anal. 29, 360\u2013365 (2019)","journal-title":"Pattern Recognit Image Anal."},{"key":"682_CR7","first-page":"1","volume":"19","author":"Y Lv","year":"2022","unstructured":"Lv, Y., Zhang, Y., Dong, S., Yang, L., Zhang, Z., Li, Z., Hu, S.: A convex hull-based feature descriptor for learning tree species classification from ALS point clouds. IEEE Geosci. Remote Sens. Lett. 19, 1\u20135 (2022)","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"682_CR8","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1080\/02331934.2020.1830089","volume":"71","author":"AA Ribeiro","year":"2022","unstructured":"Ribeiro, A.A., Sachine, M.: On the optimal separating hyperplane for arbitrary sets: a generalization of the SVM formulation and a convex hull approach. Optimization 71, 213\u2013226 (2022)","journal-title":"Optimization"},{"key":"682_CR9","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/s40314-021-01482-x","volume":"40","author":"X Zhang","year":"2021","unstructured":"Zhang, X., Wang, C., Fan, X.: Convex hull-based distance metric learning for image classification. Comput. Appl. Math. 40, 113 (2021)","journal-title":"Comput. Appl. Math."},{"key":"682_CR10","doi-asserted-by":"publisher","first-page":"4481","DOI":"10.1109\/TCSVT.2019.2926165","volume":"30","author":"H Cevikalp","year":"2020","unstructured":"Cevikalp, H., Yavuz, H.S., Triggs, B.: Face recognition based on videos by using convex hulls. IEEE Trans. Circuits Syst. Video Technol. 30, 4481\u20134495 (2020)","journal-title":"IEEE Trans. Circuits Syst. Video Technol."},{"key":"682_CR11","doi-asserted-by":"crossref","unstructured":"Gainanov, D. N., Chernavin, P. F., Rasskazova, V. A., Chernavin, N. P.: Convex hulls in solving multiclass pattern recognition problem, Learning and Intelligent Optimization, pp. 390\u2013401, Lecture Notes in Comput. Sci., 12096, Springer, Cham (2020)","DOI":"10.1007\/978-3-030-53552-0_35"},{"key":"682_CR12","doi-asserted-by":"crossref","unstructured":"Nemirko, A.: Image recognition algorithms based on the representation of classes by convex hulls, Pattern Recognition, pp. 44\u201350, Lecture Notes in Comput. Sci., 12665, Springer, Cham (2021)","DOI":"10.1007\/978-3-030-68821-9_4"},{"key":"682_CR13","doi-asserted-by":"publisher","first-page":"274","DOI":"10.1364\/OE.10.000274","volume":"10","author":"M Park","year":"2002","unstructured":"Park, M., Park, C., Park, M., Lee, C.: Algorithm for detecting human faces based on convex-hull. Opt. Express 10, 274\u2013279 (2002)","journal-title":"Opt. Express"},{"key":"682_CR14","doi-asserted-by":"crossref","unstructured":"Roy, P. P., Pal, U., Llados, J., Kimura, F.: Convex hull based approach for multi-oriented character recognition from graphical documents, In: Proceedings of the 19th International Conference on Pattern Recognition, Tampa, FL, USA, 08-11 December 2008, pp. 1\u20134 (2008)","DOI":"10.1109\/ICPR.2008.4761447"},{"key":"682_CR15","doi-asserted-by":"crossref","unstructured":"Wani, K., Ramya, S.: Hand gesture recognition using convex hull-based approach. In: S. Dhar, S. C. Mukhopadhyay, S. N. Sur, C. Liu (eds), Advances in Communication, Devices and Networking. Lecture Notes in Electrical Engineering, 776, Springer, Singapore (2022)","DOI":"10.1007\/978-981-16-2911-2_17"},{"key":"682_CR16","doi-asserted-by":"crossref","unstructured":"Wen, C., Guo, T.: An efficient algorithm for fingerprint matching based on convex hulls, Proceedings of the 2009 International Conference on Computational Intelligence and Natural Computing, Wuhan, China, pp. 66\u201369 (2009)","DOI":"10.1109\/CINC.2009.127"},{"key":"682_CR17","doi-asserted-by":"crossref","unstructured":"Zhou, X., Shi, Y.: Nearest neighbor convex hull classification method for face recognition. In: G. Allen, J. Nabrzyski, E. Seidel, G. D. Albada, J. Dongarra, P. M. A. Slootn (eds), Proceedings of the 9th International Conference on Computational Science-ICCS 2009, pp. 570\u2013577, Lecture Notes in Comput. Sci., 5545, Springer, Berlin (2009)","DOI":"10.1007\/978-3-642-01973-9_64"},{"key":"682_CR18","first-page":"1843","volume":"2","author":"G Singh","year":"2013","unstructured":"Singh, G., Swarnalatha, P., Tripathy, B.K., Kakani, S.: Convex hull based WBC computation for leukemia detection. Int. J. Adv. Res. Electr. Electron. Instrument. Eng. 2, 1843\u20131847 (2013)","journal-title":"Int. J. Adv. Res. Electr. Electron. Instrument. Eng."},{"key":"682_CR19","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1089\/cmb.2018.0216","volume":"26","author":"Y Wang","year":"2019","unstructured":"Wang, Y., Tian, K., Yau, S.S.T.: Protein sequence classification using natural vector and convex hull method. J. Comput. Biol. 26, 315\u2013321 (2019)","journal-title":"J. Comput. Biol."},{"key":"682_CR20","volume":"2369","author":"T Yan","year":"2022","unstructured":"Yan, T., Deng, Y., Chen, Y., Xia, T., Wang, D.: Nearest neighbor convex hull for health indicator construction. J. Phys: Conf. Ser. 2369, 012095 (2022)","journal-title":"J. Phys: Conf. Ser."},{"key":"682_CR21","doi-asserted-by":"publisher","first-page":"1777","DOI":"10.1016\/j.ygeno.2018.11.033","volume":"111","author":"X Zhao","year":"2019","unstructured":"Zhao, X., Tian, K., He, R.L., Yau, S.S.T.: Convex hull principle for classification and phylogeny of eukaryotic proteins. Genomics 111, 1777\u20131784 (2019)","journal-title":"Genomics"},{"key":"682_CR22","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1016\/0020-0190(72)90045-2","volume":"1","author":"RL Graham","year":"1972","unstructured":"Graham, R.L.: An efficient algorithm for determining the convex hull of a finite planar set. Inform. Process. Lett. 1, 132\u2013133 (1972)","journal-title":"Inform. Process. Lett."},{"key":"682_CR23","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1016\/0020-0190(79)90072-3","volume":"9","author":"AM Andrew","year":"1979","unstructured":"Andrew, A.M.: Another efficient algorithm for convex hulls in two dimensions. Info. Proc. Lett. 9, 216\u2013219 (1979)","journal-title":"Info. Proc. Lett."},{"key":"682_CR24","first-page":"3","volume":"34","author":"PT An","year":"2007","unstructured":"An, P.T.: A modification of Graham\u2019s algorithm for determining the convex hull of a finite planar set. Ann. Math. Inform. 34, 3\u20138 (2007)","journal-title":"Ann. Math. Inform."},{"key":"682_CR25","volume":"397","author":"PT An","year":"2021","unstructured":"An, P.T., Huyen, P.T.T., Le, N.T.: A modified Graham\u2019s convex hull algorithm for finding the connected orthogonal convex hull of a finite planar point set. Appl. Math. Comput. 397, 125889 (2021)","journal-title":"Appl. Math. Comput."},{"key":"682_CR26","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1145\/321556.321564","volume":"17","author":"DR Chand","year":"1970","unstructured":"Chand, D.R., Kapur, S.S.: An algorithm for convex polytopes. J. Assoc. Comput. Mach. 17, 78\u201386 (1970)","journal-title":"J. Assoc. Comput. Mach."},{"key":"682_CR27","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1016\/0020-0190(73)90020-3","volume":"2","author":"RA Jarvis","year":"1973","unstructured":"Jarvis, R.A.: On the identification of the convex hull of a finite set of points in the plane. Inform. Process. Lett. 2, 18\u201321 (1973)","journal-title":"Inform. Process. Lett."},{"key":"682_CR28","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1137\/0220016","volume":"20","author":"H Edelsbrunner","year":"1991","unstructured":"Edelsbrunner, H., Shi, W.: An $$O(n \\log ^2 h)$$ time algorithm for the three-dimensional convex hull problem. SIAM J. Comput. 20, 259\u2013269 (1991)","journal-title":"SIAM J. Comput."},{"key":"682_CR29","doi-asserted-by":"publisher","first-page":"1499","DOI":"10.1007\/s11075-020-00873-1","volume":"85","author":"PT An","year":"2020","unstructured":"An, P.T., Hoang, N.-D., Linh, N.K.: An efficient improvement of gift wrapping algorithm for computing the convex hull of a finite set of points in $$\\mathbb{R} ^n$$. Numer. Algorithms 85, 1499\u20131518 (2020)","journal-title":"Numer. Algorithms"},{"key":"682_CR30","doi-asserted-by":"publisher","first-page":"398","DOI":"10.1145\/355759.355766","volume":"3","author":"WF Eddy","year":"1977","unstructured":"Eddy, W.F.: A new convex hull algorithm for planar sets. ACM Trans. Math. Softw. 3, 398\u2013403 (1977)","journal-title":"ACM Trans. Math. Softw."},{"key":"682_CR31","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1016\/0020-0190(78)90021-2","volume":"7","author":"A Bykat","year":"1978","unstructured":"Bykat, A.: Convex hull of a finite set of points in two dimensions. Inf. Process. Lett. 7, 296\u2013298 (1978)","journal-title":"Inf. Process. Lett."},{"key":"682_CR32","doi-asserted-by":"crossref","unstructured":"Hoang, N.-D., Linh, N. K.: Quicker than Quickhull, Vietnam J. Math. pp. 57\u201370","DOI":"10.1007\/s10013-014-0067-1"},{"key":"682_CR33","first-page":"12462","volume":"363","author":"NK Linh","year":"2019","unstructured":"Linh, N.K., Song, C., Ryu, J., An, P.T., Hoang, N.-D., Kim, D.-S.: QuickhullDisk: A faster convex hull algorithm for disks. Appl. Math. Comput. 363, 12462 (2019)","journal-title":"Appl. Math. Comput."},{"key":"682_CR34","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1145\/359423.359430","volume":"20","author":"FP Preparata","year":"1977","unstructured":"Preparata, F.P., Hong, S.J.: Convex hulls of finite sets of points in two and three dimensions. Commun. ACM 20, 87\u201393 (1977)","journal-title":"Commun. ACM"},{"key":"682_CR35","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1137\/0215021","volume":"15","author":"DG Kirkpatrick","year":"1986","unstructured":"Kirkpatrick, D.G., Seidel, R.: The ultimate planar convex hull algorithm? SIAM J. Comput. 15, 287\u2013299 (1986)","journal-title":"SIAM J. Comput."},{"key":"682_CR36","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1016\/0020-0190(78)90003-0","volume":"7","author":"SG Akl","year":"1978","unstructured":"Akl, S.G., Toussaint, G.T.: A fast convex hull algorithm. Inform. Process. Lett. 7, 219\u2013222 (1978)","journal-title":"Inform. Process. Lett."},{"key":"682_CR37","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/0020-0190(84)90084-X","volume":"19","author":"M Kallay","year":"1984","unstructured":"Kallay, M.: The complexity of incremental convex hull algorithms in $$\\mathbb{R} ^d$$. Inform. Process. Lett. 19, 197 (1984)","journal-title":"Inform. Process. Lett."},{"key":"682_CR38","doi-asserted-by":"publisher","first-page":"466","DOI":"10.1137\/0221031","volume":"21","author":"JH Reif","year":"1992","unstructured":"Reif, J.H., Sen, S.: Optimal parallel randomized algorithms for three-dimensional convex hulls and related problems. SIAM J. Comput. 21, 466\u2013485 (1992)","journal-title":"SIAM J. Comput."},{"key":"682_CR39","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1080\/02331930802434732","volume":"59","author":"PT An","year":"2010","unstructured":"An, P.T.: Method of orienting curves for determining the convex hull of a finite set of points in the plane. Optimization 59, 175\u2013179 (2010)","journal-title":"Optimization"},{"key":"682_CR40","doi-asserted-by":"publisher","first-page":"975","DOI":"10.1080\/02331934.2011.623163","volume":"62","author":"PT An","year":"2013","unstructured":"An, P.T., Trang, L.H.: An efficient convex hull algorithm for finite point sets in 3D based on the method of orienting curves. Optimization 62, 975\u2013988 (2013)","journal-title":"Optimization"},{"key":"682_CR41","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1137\/12089702X","volume":"43","author":"KL Clarkson","year":"2014","unstructured":"Clarkson, K.L., Mulzer, W., Seshadhri, C.: Self-improving algorithms for coordinatewise maxima and convex hulls. SIAM J. Comput. 43, 617\u2013653 (2014)","journal-title":"SIAM J. Comput."},{"key":"682_CR42","doi-asserted-by":"crossref","unstructured":"Barbay, J., Ochoa, C.: Synergistic solutions for merging and computing planar convex hulls, Computing and combinatorics, pp. 156\u2013167, Lecture Notes in Comput. Sci., 10976, Springer, Cham (2018)","DOI":"10.1007\/978-3-319-94776-1_14"},{"key":"682_CR43","doi-asserted-by":"crossref","unstructured":"Ghosh, A., Schwartz, S.: Exact and approximate map-reduce algorithms for convex hull, Lecture Notes in Comput. Sci., 11346, pp. 480\u2013494. Springer, Cham (2018)","DOI":"10.1007\/978-3-030-04651-4_32"},{"key":"682_CR44","doi-asserted-by":"crossref","unstructured":"Gamby, A. N., Katajainen, J.: A faster convex-hull algorithm via bucketing, Lecture Notes in Comput. Sci., vol 11544, pp. 473\u2013489. Springer, Cham (2019)","DOI":"10.1007\/978-3-030-34029-2_30"},{"key":"682_CR45","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2019.06.014","volume":"364","author":"H Ferrada","year":"2020","unstructured":"Ferrada, H., Navarro, C.A., Hitschfeld, N.: A filtering technique for fast convex hull construction in $$\\mathbb{R} ^2$$. J. Comput. Appl. Math. 364, 112298 (2020)","journal-title":"J. Comput. Appl. Math."},{"key":"682_CR46","doi-asserted-by":"publisher","DOI":"10.1088\/1572-9494\/ac1da0","volume":"73","author":"C Wang","year":"2021","unstructured":"Wang, C., Zhou, R.-G.: A quantum search algorithm of two-dimensional convex hull. Commun. Theor. Phys. (Beijing) 73, 115102 (2021)","journal-title":"Commun. Theor. Phys. (Beijing)"},{"key":"682_CR47","volume":"481","author":"N-D Hoang","year":"2024","unstructured":"Hoang, N.-D., Linh, N.K., Phu, H.X.: Octagonal and hexadecagonal cut algorithms - linear algorithms for finding the convex hull of finite sets. Appl. Math. Comput. 481, 128931 (2024)","journal-title":"Appl. Math. Comput."},{"key":"682_CR48","doi-asserted-by":"crossref","unstructured":"Nagai, T., Yasutome, S., Tokura, N.: Convex hull problem with imprecise input, Discrete and computational geometry (Tokyo, 1998), pp. 207\u2013219, Lecture Notes in Comput. Sci., 1763, Springer, Berlin (2000)","DOI":"10.1007\/978-3-540-46515-7_18"},{"key":"682_CR49","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1016\/j.jda.2008.04.002","volume":"6","author":"M van Kreveld","year":"2008","unstructured":"van Kreveld, M., L\u00f6ffler, M.: Approximating largest convex hulls for imprecise points. J. Discrete Algorithms 6, 583\u2013594 (2008)","journal-title":"J. Discrete Algorithms"},{"key":"682_CR50","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1007\/s00453-008-9174-2","volume":"56","author":"M L\u00f6ffler","year":"2010","unstructured":"L\u00f6ffler, M., van Kreveld, M.: Largest and smallest convex hulls for imprecise points. Algorithmica 56, 235\u2013269 (2010)","journal-title":"Algorithmica"},{"key":"682_CR51","doi-asserted-by":"publisher","first-page":"3729","DOI":"10.1007\/s00500-022-06787-5","volume":"26","author":"P Birzhandi","year":"2022","unstructured":"Birzhandi, P., Kim, K.T., Youn, H.Y.: Reduction of training data for support vector machine: a survey. Soft. Comput. 26, 3729\u20133742 (2022)","journal-title":"Soft. Comput."},{"key":"682_CR52","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1016\/j.knosys.2019.02.024","volume":"173","author":"Z He","year":"2019","unstructured":"He, Z., Cheng, J., Li, J., Yang, Y.: Linear maximum margin tensor classification based on flexible convex hulls for fault diagnosis of rolling bearings. Knowl. Based Syst. 173, 62\u201373 (2019)","journal-title":"Knowl. Based Syst."},{"key":"682_CR53","unstructured":"Yousefzadeh,R.: Deep learning generalization and the convex hull of training sets, Jan. 2021, arXiv preprint arXiv:2101.09849, https:\/\/doi.org\/10.48550\/arXiv.2101.09849"},{"key":"682_CR54","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1145\/358315.358392","volume":"25","author":"JL Bentley","year":"1982","unstructured":"Bentley, J.L., Preparata, F.P., Faust, M.G.: Approximation algorithms for convex hulls. Commun. ACM 25, 64\u201368 (1982)","journal-title":"Commun. ACM"},{"key":"682_CR55","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1016\/0020-0190(83)90062-5","volume":"16","author":"E Soisalon-Soininen","year":"1983","unstructured":"Soisalon-Soininen, E.: On computing approximate convex hulls. Inform. Process. Lett. 16, 121\u2013126 (1983)","journal-title":"Inform. Process. Lett."},{"key":"682_CR56","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1016\/0167-8655(83)90017-X","volume":"2","author":"R Klette","year":"1983","unstructured":"Klette, R.: On the approximation of convex hulls of finite grid point sets. Pattern Recognit. Lett. 2, 19\u201322 (1983)","journal-title":"Pattern Recognit. Lett."},{"key":"682_CR57","unstructured":"\u017duni\u0107, J.: An outer approximation of the convex hull for finite grid, Univ. u Novom Sadu, Zb. Rad. Prirod. Mat. Fak., Ser. Mat., 2, 177\u2013185 (1992)"},{"key":"682_CR58","first-page":"597","volume":"14","author":"CE Kim","year":"1995","unstructured":"Kim, C.E., Stojmenovic, I.: Sequential and parallel approximate convex hull algorithms. Comput. Artif. Intell. 14, 597\u2013610 (1995)","journal-title":"Comput. Artif. Intell."},{"key":"682_CR59","first-page":"193","volume":"94","author":"Z-B Xu","year":"1998","unstructured":"Xu, Z.-B., Zhang, J.-S., Leung, Y.-W.: An approximate algorithm for computing multidimensional convex hulls. Appl. Math. Comput. 94, 193\u2013226 (1998)","journal-title":"Appl. Math. Comput."},{"key":"682_CR60","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/BF01758851","volume":"8","author":"Z Li","year":"1992","unstructured":"Li, Z., Milenkovic, V.: Constructing strongly convex hulls using exact or rounded arithmetic. Algorithmica 8, 345\u2013364 (1992)","journal-title":"Algorithmica"},{"key":"682_CR61","doi-asserted-by":"crossref","unstructured":"Chen, W., Deng, X. W., Wada, K., Kawaguchi, K.: Constructing a strongly convex superhull of points. Lecture Notes in Comput. Sci., 1276, pp. 42\u201351. Springer, Berlin (1997)","DOI":"10.1007\/BFb0045071"},{"key":"682_CR62","unstructured":"Hoang, N.-D., Linh, N. K., Phu, H. X.: Outer $$\\delta $$-approximation of the convex hull. In preparation"},{"key":"682_CR63","unstructured":"https:\/\/blogs.sas.com\/content\/iml\/2020\/10\/21\/random-points-in-polygon.html"},{"key":"682_CR64","unstructured":"http:\/\/www.qhull.org\/"},{"key":"682_CR65","unstructured":"https:\/\/doc.cgal.org\/latest\/Manual\/packages.html#PartConvexHullAlgorithms"},{"key":"682_CR66","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1007\/BF02237955","volume":"26","author":"L Devroye","year":"1981","unstructured":"Devroye, L., Toussaint, G.T.: A note on linear expected time algorithms for finding convex hulls. Computing 26, 361\u2013366 (1981)","journal-title":"Computing"},{"key":"682_CR67","unstructured":"https:\/\/github.com\/JuliaPolyhedra\/QHull.jl"},{"key":"682_CR68","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/0020-0190(87)90086-X","volume":"25","author":"A Melkman","year":"1987","unstructured":"Melkman, A.: On-line construction of the convex hull of a simple polyline. Inform. Process. Lett. 25, 11\u201312 (1987)","journal-title":"Inform. Process. Lett."},{"key":"682_CR69","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1007\/s41781-023-00104-x","volume":"7","author":"J Eschle","year":"2023","unstructured":"Eschle, J., Gal, T., Giordano, M., Gras, P., Hegner, B., Heinrich, L., Acosta, U.H., Kluth, S., Ling, J., Mato, P., Mikhasenko, M., Briceno, A.M., Pivarski, J., Samaras-Tsakiris, K., Schulz, O., Stewart, G.A., Strube, J., Vassilev, V.: Potential of the Julia programming language for high energy physics computing. Comput. Softw. Big. Sci. 7, 10 (2023)","journal-title":"Comput. Softw. Big. Sci."},{"key":"682_CR70","unstructured":"https:\/\/github.com\/rgcv\/CGAL.jl"}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00682-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-025-00682-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00682-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T23:01:59Z","timestamp":1749510119000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-025-00682-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,23]]},"references-count":70,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["682"],"URL":"https:\/\/doi.org\/10.1007\/s10589-025-00682-z","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"type":"print","value":"0926-6003"},{"type":"electronic","value":"1573-2894"}],"subject":[],"published":{"date-parts":[[2025,4,23]]},"assertion":[{"value":"14 October 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 March 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 April 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}