Content deleted Content added
|state=collapsed — we do not need to spam a long list of undigested and unintelligible jargon at the ends of articles in their default view |
|||
| (30 intermediate revisions by 17 users not shown) | |||
Line 3:
{{good article}}
[[File:Extreme points.svg|thumb|right|The convex hull of the red set is the blue and red [[convex set]].]]
In [[geometry]], the '''convex hull'''
Convex hulls of [[open set]]s are open, and convex hulls of [[compact set]]s are compact. Every compact convex set is the convex hull of its [[extreme point]]s. The convex hull operator is an example of a [[closure operator]], and every [[antimatroid]] can be represented by applying this closure operator to finite sets of points.
Line 44:
===Preservation of topological properties===
[[File:Versiera007.svg|thumb|The [[witch of Agnesi]]. The points on or above the red curve provide an example of a closed set whose convex hull is open (the open [[upper half-plane]]).]]
Topologically, the convex hull of an [[open set]] is always itself open, and (in Euclidean spaces) the convex hull of a compact set is always itself compact. However, there exist closed sets for which the convex hull is not closed.<ref>{{harvtxt|Grünbaum|2003}}, p. 16; {{harvtxt|Lay|1982}}, p. 21; {{harvtxt|Sakuma|1977}}.</ref> For instance, the closed set
:<math>\left \{ (x,y) \mathop{\bigg|} y\ge \frac{1}{1+x^2}\right\}</math>
Line 50:
(the set of points that lie on or above the [[witch of Agnesi]]) has the open [[upper half-plane]] as its convex hull.<ref>This example is given by {{harvtxt|Talman|1977}}, Remark 2.6.</ref>
===Extreme points===
Line 84:
===Simple polygons===
{{main|Convex hull of a simple polygon}}
[[File:Convex hull of a simple polygon.svg|thumb|upright|Convex hull (
The convex hull of a [[simple polygon]] encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions, bounded by a [[polygonal chain]] of the polygon and a single convex hull edge, are called ''pockets''. Computing the same decomposition recursively for each pocket forms a hierarchical description of a given polygon called its ''convex differences tree''.{{sfnp|Rappoport|1992}} Reflecting a pocket across its convex hull edge expands the given simple polygon into a polygon with the same perimeter and larger area, and the [[Erdős–Nagy theorem]] states that this expansion process eventually terminates.{{sfnp|Demaine|Gassend|O'Rourke|Toussaint|2008}}
Line 108:
[[Dynamic convex hull]] data structures can be used to keep track of the convex hull of a set of points undergoing insertions and deletions of points,{{sfnp|Chan|2012}} and [[kinetic convex hull]] structures can keep track of the convex hull for points moving continuously.{{sfnp|Basch|Guibas|Hershberger|1999}}
The construction of convex hulls also serves as a tool, a building block for a number of other computational-geometric algorithms such as the [[rotating calipers]] method for computing the [[width]] and [[Diameter (computational geometry)|diameter]] of a point set.{{sfnp|Toussaint|1983}}
== Related structures ==
Line 137:
== Applications ==
[[File:CIE1931xy_gamut_comparison.svg|thumb|The convex hull of the primary colors in each [[color space]] on a [[CIE 1931]] xy [[chromaticity diagram]] defines the space's [[gamut]] of possible colors]]
Convex hulls have wide applications in many fields. Within mathematics, convex hulls are used to study [[polynomial]]s, matrix [[eigenvalue]]s, and [[unitary element]]s, and several theorems in [[discrete geometry]] involve convex hulls. They are used in [[robust statistics]] as the outermost contour of [[Tukey depth]], are part of the [[bagplot]] visualization of two-dimensional data, and define risk sets of [[randomised decision rule|randomized decision rule]]s. Convex hulls of [[indicator vector]]s of solutions to combinatorial problems are central to [[combinatorial optimization]] and [[polyhedral combinatorics]]. In economics, convex hulls can be used to apply methods of [[convexity in economics]] to non-convex markets. In geometric modeling, the convex hull property [[Bézier curve]]s helps find their crossings, and convex hulls are part of the measurement of boat hulls. And in the study of animal behavior, convex hulls are used in a standard definition of the [[home range]].
===Mathematics===
[[Newton polygon]]s of univariate [[polynomial]]s and [[Newton polytope]]s of multivariate polynomials are convex hulls of points derived from the exponents of the terms in the polynomial, and can be used to analyze the [[asymptotic analysis|asymptotic]] behavior of the polynomial and the valuations of its roots.<ref>{{harvtxt|Artin|1967}}; {{harvtxt|Gel'fand|Kapranov|Zelevinsky|1994}}</ref> Convex hulls and polynomials also come together in the [[Gauss–Lucas theorem]], according to which the [[Zero of a function|roots]] of the derivative of a polynomial all lie within the convex hull of the roots of the polynomial.{{sfnp|Prasolov|2004}}
[[File:Tverberg heptagon.svg|thumb|upright|Partition of seven points into three subsets with intersecting convex hulls, guaranteed to exist for any seven points in the plane by [[Tverberg's theorem]]]]
In [[Spectral theory|spectral analysis]], the [[numerical range]] of a [[normal matrix]] is the convex hull of its [[eigenvalue]]s.{{sfnp|Johnson|1976}}
The [[Russo–Dye theorem]] describes the convex hulls of [[unitary element]]s in a [[C*-algebra]].{{sfnp|Gardner|1984}}
Line 189 ⟶ 190:
==References==
{{refbegin|30em}}
*{{citation |last=Fan |first=Ky |title=Convex Sets and Their Applications. Summer Lectures 1959. |url=https://books.google.com/books?id=QKkrAAAAYAAJ&pg=PA48 |publisher=Argon national laboratory |year=1959}}
*{{citation |last=Andrew |first=A. M. |doi=10.1016/0020-0190(79)90072-3 |issue=5 |journal=[[Information Processing Letters]] |pages=216–219 |title=Another efficient algorithm for convex hulls in two dimensions |volume=9 |year=1979}}
*{{citation |last=Artin |first=Emil |author-link=Emil Artin |contribution=2.5. Newton's Polygon |contribution-url=https://books.google.com/books?id=VixOGTdZaCQC&pg=PA37 |mr=0237460 |pages=37–43 |publisher=Gordon and Breach |title=Algebraic Numbers and Algebraic Functions |year=1967}}
*{{citation |last=Auel |first=Asher |issue=3 |journal=[[Notices of the American Mathematical Society]] |mr=3889348 |pages=330–340 |title=The mathematics of Grace Murray Hopper |url=https://www.ams.org/journals/notices/201903/rnoti-p330.pdf |volume=66 |year=2019 |doi=10.1090/noti1810 |s2cid=76650751}}
*{{citation |last1=Avis |first1=David |author1-link=David Avis |last2=Bremner |first2=David |last3=Seidel |first3=Raimund |author3-link=Raimund Seidel |doi=10.1016/S0925-7721(96)00023-5 |issue=5–6 |journal=[[Computational Geometry (journal)|Computational Geometry]] |mr=1447243 |pages=265–301 |title=How good are convex hull algorithms? |volume=7 |year=1997}}
*{{citation |last1=Bárány |first1=Imre |author1-link=Imre Bárány |last2=Katchalski |first2=Meir |last3=Pach |first3=János |author3-link=János Pach |doi=10.1090/S0002-9939-1982-0663877-X |doi-access=free |issue=1 |journal=[[Proceedings of the American Mathematical Society]] |mr=663877 |pages=109–114 |title=Quantitative Helly-type theorems |volume=86 |year=1982 |jstor=2044407}}
*{{citation |last1=Basch |first1=Julien |last2=Guibas |first2=Leonidas J. |author2-link=Leonidas J. Guibas |last3=Hershberger |first3=John |author3-link=John Hershberger |citeseerx=10.1.1.134.6921 |doi=10.1006/jagm.1998.0988 |issue=1 |journal=[[Journal of Algorithms]] |mr=1670903 |pages=1–28 |title=Data structures for mobile data |volume=31 |year=1999 |s2cid=8013433}}
*{{citation |last=Birkhoff |first=Garrett |author-link=Garrett Birkhoff |doi=10.2307/1989687 |issue=2 |journal=[[Transactions of the American Mathematical Society]] |mr=1501815 |pages=357–378 |title=Integration of functions with values in a Banach space |volume=38 |year=1935 |jstor=1989687}}
*{{citation |last=Brown |first=K. Q. |doi=10.1016/0020-0190(79)90074-7 |issue=5 |journal=[[Information Processing Letters]] |pages=223–228 |title=Voronoi diagrams from convex hulls |volume=9 |year=1979 |s2cid=44537056}}
*{{citation |last1=de Berg |first1=M. |author1-link=Mark de Berg |last2=van Kreveld |first2=M. |author2-link=Marc van Kreveld |last3=Overmars |first3=Mark |author3-link=Mark Overmars |last4=Schwarzkopf |first4=O. |author4-link=Otfried Cheong |edition=3rd |publisher=Springer |title=Computational Geometry: Algorithms and Applications |year=2008}}
*{{citation |last=Chan |first=Timothy M. |author-link=Timothy M. Chan |doi=10.1142/S0218195912600096 |issue=4 |journal=[[International Journal of Computational Geometry and Applications]] |mr=2994585 |pages=341–364 |title=Three problems about dynamic convex hulls |volume=22 |year=2012}}
*{{citation |last1=Chang |first1=J. S. |last2=Yap |first2=C.-K. |doi=10.1007/BF02187692 |issue=2 |journal=[[Discrete & Computational Geometry]] |mr=834056 |pages=155–182 |title=A polynomial solution for the potato-peeling problem |volume=1 |year=1986 |doi-access=free}}
*{{citation |last=Chazelle |first=Bernard |author-link=Bernard Chazelle |doi=10.1109/TIT.1985.1057060 |issue=4 |journal=[[IEEE Transactions on Information Theory]] |mr=798557 |pages=509–517 |title=On the convex layers of a planar set |volume=31 |year=1985}}
*{{citation |last=Chazelle |first=Bernard |author-link=Bernard Chazelle |citeseerx=10.1.1.113.8709 |doi=10.1007/BF02573985 |issue=1 |journal=[[Discrete & Computational Geometry]] |pages=377–409 |title=An optimal convex hull algorithm in any fixed dimension |url=https://www.cs.princeton.edu/~chazelle/pubs/ConvexHullAlgorithm.pdf |volume=10 |year=1993 |s2cid=26605267}}
*{{citation |last1=Chen |first1=Qinyu |last2=Wang |first2=Guozhao |date=March 2003 |doi=10.1016/s0167-8396(03)00003-7 |issue=1 |journal=Computer Aided Geometric Design |pages=29–39 |title=A class of Bézier-like curves |volume=20}}
*{{citation |last1=Cranston |first1=M. |last2=Hsu |first2=P. |last3=March |first3=P. |issue=1 |journal=[[Annals of Probability]] |jstor=2244202 |mr=972777 |pages=144–150 |title=Smoothness of the convex hull of planar Brownian motion |volume=17 |year=1989 |doi=10.1214/aop/1176991500 |doi-access=free}}
*{{citation |last1=Demaine |first1=Erik D. |author1-link=Erik Demaine |last2=Gassend |first2=Blaise |last3=O'Rourke |first3=Joseph |author3-link=Joseph O'Rourke (professor) |last4=Toussaint |first4=Godfried T. |author4-link=Godfried Toussaint |contribution=All polygons flip finitely ... right? |doi=10.1090/conm/453/08801 |___location=Providence, Rhode Island |mr=2405683 |pages=231–255 |publisher=American Mathematical Society |series=Contemporary Mathematics |title=Surveys on Discrete and Computational Geometry |volume=453 |year=2008 |isbn=978-0-8218-4239-3}}
*{{citation |last=Dines |first=L. L. |author-link=Lloyd Dines |doi=10.2307/2302604 |issue=4 |journal=[[American Mathematical Monthly]] |jstor=2302604 |mr=1524247 |pages=199–209 |title=On convexity |volume=45 |year=1938}}
*{{citation |last1=Dirnböck |first1=Hans |last2=Stachel |first2=Hellmuth |author2-link=Hellmuth Stachel |issue=2 |journal=Journal for Geometry and Graphics |mr=1622664 |pages=105–118 |title=The development of the oloid |url=http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0113.pdf |volume=1 |year=1997}}
*{{citation |last1=Edelsbrunner |first1=Herbert |author1-link=Herbert Edelsbrunner |last2=Kirkpatrick |first2=David G. |author2-link=David G. Kirkpatrick |last3=Seidel |first3=Raimund |author3-link=Raimund Seidel |doi=10.1109/TIT.1983.1056714 |issue=4 |journal=[[IEEE Transactions on Information Theory]] |pages=551–559 |title=On the shape of a set of points in the plane |volume=29 |year=1983}}
*{{citation |last1=Epstein |first1=D. B. A. |author1-link=David B. A. Epstein |last2=Marden |first2=A. |author2-link=Albert Marden |editor-last=Epstein |editor-first=D. B. A. |editor-link=David B. A. Epstein |contribution=Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces |mr=903852 |pages=113–253 |publisher=Cambridge University Press |___location=Cambridge |series=London Mathematical Society Lecture Note Series |title=Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) |volume=111 |year=1987}}
*{{citation |last1=Escobar |first1=Laura |last2=Kaveh |first2=Kiumars |date=September 2020 |issue=8 |journal=Notices of the American Mathematical Society |pages=1116–1123 |title=Convex polytopes, algebraic geometry, and combinatorics |url=https://www.ams.org/journals/notices/202008/rnoti-p1116.pdf |volume=67 |doi=10.1090/noti2137 |s2cid=221659506}}
*{{citation |last=Fultz |first=Brent |date=April 2020 |doi=10.1017/9781108641449 |page=55 |publisher=Cambridge University Press |title=Phase Transitions in Materials |isbn=9781108641449 |url=https://books.google.com/books?id=AkbhDwAAQBAJ&pg=PA55}}
*{{citation |last=Gardner |first=L. Terrell |doi=10.2307/2044692 |issue=1 |journal=[[Proceedings of the American Mathematical Society]] |mr=722439 |page=171 |title=An elementary proof of the Russo-Dye theorem |volume=90 |year=1984 |jstor=2044692 |s2cid=119501393}}
*{{citation |last1=Gel'fand |first1=I. M. |author1-link=Israel Gelfand |last2=Kapranov |first2=M. M. |author2-link=Mikhail Kapranov |last3=Zelevinsky |first3=A. V. |author3-link=Andrei Zelevinsky |contribution=6. Newton Polytopes and Chow Polytopes |doi=10.1007/978-0-8176-4771-1 |isbn=0-8176-3660-9 |mr=1264417 |pages=193–213 |publisher=Birkhäuser |series=Mathematics: Theory & Applications |title=Discriminants, Resultants, and Multidimensional Determinants |year=1994}}
*{{citation |last1=Getz |first1=Wayne M. |last2=Wilmers |first2=Christopher C. |doi=10.1111/j.0906-7590.2004.03835.x |issue=4 |journal=[[Ecography]] |pages=489–505 |publisher=Wiley |title=A local nearest-neighbor convex-hull construction of home ranges and utilization distributions |url=https://www.cnr.berkeley.edu/~getz/Reprints04/Getz&WilmersEcoG_SF_04.pdf |volume=27 |year=2004 |bibcode=2004Ecogr..27..489G |s2cid=14592779}}
*{{citation |last=Gibbs |first=Willard J. |author-link=Josiah Willard Gibbs |journal=Transactions of the Connecticut Academy of Arts and Sciences |pages=382–404 |title=A method of geometrical representation of the thermodynamic properties of substances by means of surfaces |volume=2 |year=1873}}; reprinted in ''[https://archive.org/details/scientificpaper00gibbgoog The Scientific Papers of J. Willard Gibbs, Vol. I: Thermodynamics]'', Longmans, Green, & Co., 1906, [https://archive.org/details/scientificpaper00gibbgoog/page/n67 pp. 33–54]
*{{citation |last1=Graham |first1=Ronald L. |author1-link=Ronald Graham |last2=Yao |first2=F. Frances |author2-link=Frances Yao |doi=10.1016/0196-6774(83)90013-5 |issue=4 |journal=[[Journal of Algorithms]] |mr=729228 |pages=324–331 |title=Finding the convex hull of a simple polygon |volume=4 |year=1983}}
*{{citation |last=Grünbaum |first=Branko |author-link=Branko Grünbaum |edition=2nd |isbn=9780387004242 |publisher=Springer |series=Graduate Texts in Mathematics |title=Convex Polytopes |title-link=Convex Polytopes |volume=221 |year=2003}}
*{{citation |last=Gustin |first=William |doi=10.1090/S0002-9904-1947-08787-5 |journal=[[Bulletin of the American Mathematical Society]] |mr=20800 |pages=299–301 |title=On the interior of the convex hull of a Euclidean set |volume=53 |year=1947 |issue=4 |doi-access=free}}
*{{citation |last=Harris |first=Bernard |contribution=Mathematical models for statistical decision theory |contribution-url=https://apps.dtic.mil/dtic/tr/fulltext/u2/737250.pdf |mr=0356305 |pages=369–389 |title=Optimizing methods in statistics (Proc. Sympos., Ohio State Univ., Columbus, Ohio, 1971) |year=1971 |access-date=2020-01-01 |archive-date=2021-02-28 |archive-url=https://web.archive.org/web/20210228072623/https://apps.dtic.mil/dtic/tr/fulltext/u2/737250.pdf |url-status=dead}}
*{{citation |last=Hautier |first=Geoffroy |editor1-last=Atahan-Evrenk |editor1-first=Sule |editor2-last=Aspuru-Guzik |editor2-first=Alan |contribution=Data mining approaches to high-throughput crystal structure and compound prediction |doi=10.1007/128_2013_486 |pages=139–179 |pmid=24287952 |publisher=Springer International Publishing |series=Topics in Current Chemistry |title=Prediction and Calculation of Crystal Structures: Methods and Applications |volume=345 |year=2014 |isbn=978-3-319-05773-6}}; see [https://books.google.com/books?id=9nu5BQAAQBAJ&pg=PA143 p. 143]
*{{citation |last=Herrlich |first=Horst |author-link=Horst Herrlich |department=Proceedings of the Symposium on General Topology and Applications (Oxford, 1989) |doi=10.1016/0166-8641(92)90092-E |issue=1–3 |journal=[[Topology and Its Applications]] |mr=1173256 |pages=181–187 |title=Hyperconvex hulls of metric spaces |volume=44 |year=1992 |doi-access=free}}
*{{citation |last=Johnson |first=Charles R. |author-link=Charles Royal Johnson |doi=10.1016/0024-3795(76)90080-x |issue=1 |journal=[[Linear Algebra and Its Applications]] |mr=460358 |pages=89–94 |title=Normality and the numerical range |volume=15 |year=1976 |doi-access=free}}
*{{citation |last1=Kashiwabara |first1=Kenji |last2=Nakamura |first2=Masataka |last3=Okamoto |first3=Yoshio |citeseerx=10.1.1.14.4965 |doi=10.1016/j.comgeo.2004.05.001 |issue=2 |journal=[[Computational Geometry (journal)|Computational Geometry]] |mr=2107032 |pages=129–144 |title=The affine representation theorem for abstract convex geometries |volume=30 |year=2005}}
*{{citation |last=Katoh |first=Naoki |journal=IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |pages=321–329 |title=Bicriteria network optimization problems |volume=E75-A |year=1992}}
*{{citation |last1=Kernohan |first1=Brian J. |last2=Gitzen |first2=Robert A. |last3=Millspaugh |first3=Joshua J. |editor1-last=Millspaugh |editor1-first=Joshua |editor2-last=Marzluff |editor2-first=John M. |contribution=Analysis of animal space use and movements |isbn=9780080540221 |publisher=Academic Press |title=Radio Tracking and Animal Populations |year=2001}}
*{{citation |last1=Kim |first1=Sooran |last2=Kim |first2=Kyoo |last3=Koo |first3=Jahyun |last4=Lee |first4=Hoonkyung |last5=Min |first5=Byung Il |last6=Kim |first6=Duck Young |bibcode=2019NatSR...920253K |date=December 2019 |doi=10.1038/s41598-019-56497-6 |issue=1 |journal=Scientific Reports |page=20253 |pmc=6934831 |pmid=31882982 |title=Pressure-induced phase transitions and superconductivity in magnesium carbides |volume=9}}
*{{citation |last=Kirkpatrick |first=K. A. |arxiv=quant-ph/0305068 |doi=10.1007/s10702-006-1852-1 |issue=1 |journal=[[Foundations of Physics Letters]] |pages=95–102 |title=The Schrödinger–HJW theorem |volume=19 |year=2006 |bibcode=2006FoPhL..19...95K |s2cid=15995449}}
*{{citation |last=Kiselman |first=Christer O. |doi=10.1090/S0002-9947-02-02915-X |issue=5 |journal=[[Transactions of the American Mathematical Society]] |mr=1881029 |pages=2035–2053 |title=A semigroup of operators in convexity theory |volume=354 |year=2002 |doi-access=free}}
*{{citation |last=Knuth |first=Donald E. |author-link=Donald Knuth |doi=10.1007/3-540-55611-7 |isbn=3-540-55611-7 |___location=Heidelberg |mr=1226891 |publisher=Springer-Verlag |series=Lecture Notes in Computer Science |title=Axioms and Hulls |url=http://www-cs-faculty.stanford.edu/~uno/aah.html |volume=606 |year=1992 |s2cid=5452191 |access-date=2011-09-15 |archive-date=2017-06-20 |archive-url=https://web.archive.org/web/20170620062425/http://www-cs-faculty.stanford.edu/~uno/aah.html |url-status=dead}}
*{{citation |last=Kőnig |first=Dénes |author-link=Dénes Kőnig |date=December 1922 |doi=10.1007/bf01215899 |issue=1 |journal=[[Mathematische Zeitschrift]] |pages=208–210 |title=Über konvexe Körper |volume=14 |s2cid=128041360}}; see also review by [[Hans Rademacher]] (1922), {{JFM|48.0835.01}}
*{{citation |last1=Krein |first1=Mark |author1-link=Mark Krein |last2=Milman |first2=David |author2-link=David Milman |journal=[[Studia Mathematica]] |pages=133–138 |title=On extreme points of regular convex sets |url=https://eudml.org/doc/219061 |volume=9 |year=1940 |doi=10.4064/sm-9-1-133-138 |doi-access=free}}
*{{citation |last1=Krein |first1=M. |author1-link=Mark Krein |last2=Šmulian |first2=V. |doi=10.2307/1968735 |journal=[[Annals of Mathematics]] |series=Second Series |jstor=1968735 |mr=2009 |pages=556–583 |title=On regularly convex sets in the space conjugate to a Banach space |volume=41 |year=1940 |issue=3 |hdl=10338.dmlcz/100106 |hdl-access=free}}
*{{citation |last=Laurentini |first=A. |doi=10.1109/34.273735 |issue=2 |journal=IEEE Transactions on Pattern Analysis and Machine Intelligence |pages=150–162 |title=The visual hull concept for silhouette-based image understanding |volume=16 |year=1994}}
*{{citation |last=Lay |first=Steven R. |isbn=0-471-09584-2 |mr=655598 |publisher=John Wiley & Sons |title=Convex Sets and their Applications |year=1982}}
*{{citation |last=Lee |first=D. T. |author-link=Der-Tsai Lee |doi=10.1007/BF00993195 |issue=2 |journal=International Journal of Computer and Information Sciences |mr=724699 |pages=87–98 |title=On finding the convex hull of a simple polygon |volume=12 |year=1983 |s2cid=28600832}}
*{{citation |last=Mason |first=Herbert B. |page=698 |title=Encyclopaedia of Ships and Shipping |url=https://books.google.com/books?id=d3gDAAAAYAAJ&pg=PA698 |year=1908}}
*{{citation |last1=McCallum |first1=Duncan |last2=Avis |first2=David |author2-link=David Avis |doi=10.1016/0020-0190(79)90069-3 |issue=5 |journal=[[Information Processing Letters]] |mr=552534 |pages=201–206 |title=A linear algorithm for finding the convex hull of a simple polygon |volume=9 |year=1979}}
*{{citation |last=Newton |first=Isaac |author-link=Isaac Newton |date=October 24, 1676 |publisher=University of Oxford |title=Letter to Henry Oldenburg |url=https://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00196 |work=The [[Newton Project]]}}
*{{citation |last=Nicola |first=Piercarlo |contribution=General Competitive Equilibrium |doi=10.1007/978-3-662-04238-0_16 |pages=197–215 |publisher=Springer |title=Mainstream Mathematical Economics in the 20th Century |year=2000 |isbn=978-3-642-08638-0}}
*{{citation |last1=Nilsen |first1=Erlend B. |last2=Pedersen |first2=Simen |last3=Linnell |first3=John D. C. |year=2008 |doi=10.1007/s11284-007-0421-9 |issue=3 |journal=Ecological Research |pages=635–639 |title=Can minimum convex polygon home ranges be used to draw biologically meaningful conclusions? |volume=23 |bibcode=2008EcoR...23..635N |s2cid=30843551}}
*{{citation |last=Oberman |first=Adam M. |doi=10.1090/S0002-9939-07-08887-9 |issue=6 |journal=[[Proceedings of the American Mathematical Society]] |mr=2286077 |pages=1689–1694 |title=The convex envelope is the solution of a nonlinear obstacle problem |volume=135 |year=2007 |doi-access=free}}
*{{citation |last=Okon |first=T. |doi=10.4171/ZAA/952 |issue=2 |journal=Zeitschrift für Analysis und ihre Anwendungen |mr=1768994 |pages=303–314 |title=Choquet theory in metric spaces |volume=19 |year=2000 |doi-access=free}}
*{{citation |last1=Ottmann |first1=T. |last2=Soisalon-Soininen |first2=E. |last3=Wood |first3=Derick |author3-link=Derick Wood |doi=10.1016/0020-0255(84)90025-2 |issue=3 |journal=[[Information Sciences (journal)|Information Sciences]] |pages=157–171 |title=On the definition and computation of rectilinear convex hulls |volume=33 |year=1984}}
*{{citation |last=Prasolov |first=Victor V. |contribution=1.2.1 The Gauss–Lucas theorem |contribution-url=https://books.google.com/books?id=b1a7ye_EjZwC&pg=PA12 |doi=10.1007/978-3-642-03980-5 |isbn=3-540-40714-6 |mr=2082772 |pages=12–13 |publisher=Springer |series=Algorithms and Computation in Mathematics |title=Polynomials |volume=11 |year=2004}}
*{{citation |last=Pulleyblank |first=W. R. |author-link=William R. Pulleyblank |editor1-last=Bachem |editor1-first=Achim |editor2-last=Korte |editor2-first=Bernhard |editor3-last=Grötschel |editor3-first=Martin |contribution=Polyhedral combinatorics |doi=10.1007/978-3-642-68874-4_13 |pages=312–345 |publisher=Springer |title=Mathematical Programming: The State of the Art (XIth International Symposium on Mathematical Programming, Bonn 1982) |year=1983 |isbn=978-3-642-68876-8}}
*{{citation |last=Rappoport |first=Ari |doi=10.1111/1467-8659.1140235 |issue=4 |journal=Computer Graphics Forum |pages=235–240 |title=An efficient adaptive algorithm for constructing the convex differences tree of a simple polygon |volume=11 |year=1992 |s2cid=20137707}}
*{{citation |last=Reay |first=John R. |doi=10.1007/BF02760885 |doi-access=free |issue=3 |journal=[[Israel Journal of Mathematics]] |mr=570883 |pages=238–244 (1980) |title=Several generalizations of Tverberg's theorem |volume=34 |year=1979 |s2cid=121352925}}
*{{citation |last1=Rieffel |first1=Eleanor G. |author1-link=Eleanor Rieffel |last2=Polak |first2=Wolfgang H. |isbn=978-0-262-01506-6 |pages=215–216 |publisher=MIT Press |title=Quantum Computing: A Gentle Introduction |title-link=Quantum Computing: A Gentle Introduction |year=2011}}
*{{citation |last=Rockafellar |first=R. Tyrrell |author-link=R. Tyrrell Rockafellar |mr=0274683 |publisher=Princeton University Press |___location=Princeton, N.J. |series=Princeton Mathematical Series |title=Convex Analysis |volume=28 |year=1970}}
*{{citation |last=Rossi |first=Hugo |author-link=Hugo Rossi |doi=10.2307/1970292 |journal=[[Annals of Mathematics]] |jstor=1970292 |mr=133479 |pages=470–493 |series=Second Series |title=Holomorphically convex sets in several complex variables |volume=74 |year=1961 |issue=3}}
*{{citation |last1=Rousseeuw |first1=Peter J. |author1-link=Peter Rousseeuw |last2=Ruts |first2=Ida |last3=Tukey |first3=John W. |author3-link=John Tukey |doi=10.1080/00031305.1999.10474494 |issue=4 |journal=[[The American Statistician]] |pages=382–387 |title=The bagplot: A bivariate boxplot |volume=53 |year=1999}}
*{{citation |last=Sakuma |first=Itsuo |doi=10.1016/0022-0531(77)90095-3 |issue=1 |journal=[[Journal of Economic Theory]] |pages=223–227 |title=Closedness of convex hulls |volume=14 |year=1977}}
*{{citation |last=Schneider |first=Rolf |author-link=Rolf Schneider |doi=10.1017/CBO9780511526282 |isbn=0-521-35220-7 |___location=Cambridge |mr=1216521 |publisher=Cambridge University Press |series=Encyclopedia of Mathematics and its Applications |title=Convex Bodies: The Brunn–Minkowski Theory |url=https://archive.org/details/convexbodiesbrun0000schn |volume=44 |year=1993}}
*{{citation |last=Seaton |first=Katherine A. |arxiv=1603.08409 |doi=10.1080/17513472.2017.1318512 |issue=4 |journal=[[Journal of Mathematics and the Arts]] |mr=3765242 |pages=187–202 |title=Sphericons and D-forms: a crocheted connection |volume=11 |year=2017 |s2cid=84179479}}
*{{citation |last=Sedykh |first=V. D. |issue=6 |journal=Trudy Seminara imeni I. G. Petrovskogo |mr=630708 |pages=239–256 |title=Structure of the convex hull of a space curve |year=1981}}, translated in ''Journal of Soviet Mathematics'' 33 (4): 1140–1153, 1986, {{doi|10.1007/BF01086114}}
*{{citation |last=Sontag |first=Eduardo D. |author-link=Eduardo D. Sontag |issue=1 |journal=[[Pacific Journal of Mathematics]] |mr=644949 |pages=183–201 |title=Remarks on piecewise-linear algebra |url=https://projecteuclid.org/euclid.pjm/1102734396 |volume=98 |year=1982 |doi=10.2140/pjm.1982.98.183 |s2cid=18446330 |doi-access=free}}
*{{citation |last=Steinitz |first=E. |author-link=Ernst Steinitz |doi=10.1515/crll.1914.144.1 |journal=[[Crelle's Journal|Journal für die Reine und Angewandte Mathematik]] |mr=1580890 |pages=1–40 |title=Bedingt konvergente Reihen und konvexe Systeme. (Fortsetzung) |volume=1914 |year=1914 |issue=144 |s2cid=122998337}}
*{{citation |last=Talman |first=Louis A. |issue=1–2 |journal=Kōdai Mathematical Seminar Reports |mr=463985 |pages=62–70 |title=Fixed points for condensing multifunctions in metric spaces with convex structure |url=https://projecteuclid.org/euclid.kmj/1138833572 |volume=29 |year=1977}}
*{{citation |last=Toussaint |first=Godfried |author-link=Godfried Toussaint |citeseerx=10.1.1.155.5671 |contribution=Solving geometric problems with the rotating calipers |title=Proceedings of IEEE MELECON '83, Athens |year=1983}}
*{{citation |last=Toussaint |first=Godfried |author-link=Godfried Toussaint |contribution=An optimal algorithm for computing the relative convex hull of a set of points in a polygon |pages=853–856 |publisher=North-Holland |title=Proceedings of EURASIP, Signal Processing III: Theories and Applications, Part 2 |year=1986}}
*{{citation |last=Weeks |first=Jeffrey R. |author-link=Jeffrey Weeks (mathematician) |doi=10.1016/0166-8641(93)90032-9 |issue=2 |journal=[[Topology and Its Applications]] |mr=1241189 |pages=127–149 |title=Convex hulls and isometries of cusped hyperbolic 3-manifolds |volume=52 |year=1993 |doi-access=free}}
*{{citation |last=Westermann |first=L. R. J. |doi=10.1016/1385-7258(76)90065-2 |issue=2 |journal=[[Indagationes Mathematicae]] |mr=0404097 |pages=179–184 |title=On the hull operator |volume=38 |year=1976 |doi-access=free}}
*{{citation |last=White |first=F. Puryer |date=April 1923 |issue=68 |journal=Science Progress in the Twentieth Century |jstor=43432008 |pages=517–526 |title=Pure mathematics |volume=17}}
*{{citation |last=Whitley |first=Robert |doi=10.2307/2046536 |issue=2 |journal=[[Proceedings of the American Mathematical Society]] |mr=835903 |pages=376–377 |title=The Kreĭn-Šmulian theorem |volume=97 |year=1986 |jstor=2046536}}
*{{citation |last1=Williams |first1=Jason |last2=Rossignac |first2=Jarek |editor1-last=Kobbelt |editor1-first=Leif |editor2-last=Shapiro |editor2-first=Vadim |contribution=Tightening: curvature-limiting morphological simplification |doi=10.1145/1060244.1060257 |hdl=1853/3736 |pages=107–112 |publisher=ACM |title=Proceedings of the Tenth ACM Symposium on Solid and Physical Modeling 2005, Cambridge, Massachusetts, USA, June 13-15, 2005 |year=2005 |s2cid=15514388}}
*{{citation |last=Worton |first=Bruce J. |doi=10.2307/2533254 |issue=4 |journal=[[Biometrics (journal)|Biometrics]] |jstor=2533254 |pages=1206–1215 |title=A convex hull-based estimator of home-range size |volume=51 |year=1995}}
{{refend}}
| |||