Convex set: Difference between revisions

Revision as of 18:23, 30 October 2020

Revision as of 23:28, 1 December 2020

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[[File:Convex polygon illustration2.svg|right|thumb|Illustration of a non-convex set. Since the (red) part of the (black and red) line-segment joining the points x and y lies ''outside'' of the (green) set, the set is non-convex.]]

In [[geometry]], a subset of a [[Euclidean space]], or more generally an [[affine space]] over the [[Real number|reals]], is '''convex''' if, given any two points, it contains the whole [[line segment]] that joins them. Equivalently, a '''convex set''' or a '''convex region''' is a subset that intersect every [[line (geometry)|line]] into a single line segment (possibly empty).<ref>{{cite book|last1=Morris|first1=Carla C.|last2=Stark|first2=Robert M.|title=Finite Mathematics: Models and Applications|date=24 August 2015|publisher=John Wiley & Sons|isbn=9781119015383|page=121|url=https://books.google.com/books?id=ZgJyCgAAQBAJ&q=convex+region&pg=PA121|accessdate=5 April 2017|language=en}}</ref><ref>{{cite journal|last1=Kjeldsen|first1=Tinne Hoff|title=History of Convexity and Mathematical Programming|journal=Proceedings of the International Congress of Mathematicians|issue=ICM 2010|pages=3233–3257|doi=10.1142/9789814324359_0187|url=http://www.mathunion.org/ICM/ICM2010.4/Main/icm2010.4.3233.3257.pdf|accessdate=5 April 2017|url-status=dead|archiveurlarchive-url=https://web.archive.org/web/20170811100026/http://www.mathunion.org/ICM/ICM2010.4/Main/icm2010.4.3233.3257.pdf|archivedatearchive-date=2017-08-11}}</ref>

For example, a solid [[cube (geometry)|cube]] is a convex set, but anything that is hollow or has an indent, for example, a [[crescent]] shape, is not convex.

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=== Blaschke-Santaló diagrams ===

The set <math>\mathcal{K}^2</math> of all planar convex bodies can be parameterized in terms of the convex body [[Diameter#Generalizations|diameter]] ''D'', its inradius ''r'' (the biggest circle contained in the convex body) and its circumradius ''R'' (the smallest circle containing the convex body). In fact, this set can be described by the set of inequalities given by<ref name=":0">{{Cite journal|last=Santaló|first=L.|date=1961|title=Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa planas|url=|journal=Mathematicae Notae|volume=17|pages=82–104|via=}}</ref><ref name=":1">{{Cite journal|last1=Brandenberg|first1=René|last2=González Merino|first2=Bernardo|date=2017|title=A complete 3-dimensional Blaschke-Santaló diagram|url=http://mia.ele-math.com/20-22|journal=Mathematical Inequalities & Applications|language=en|issue=2|pages=301–348|doi=10.7153/mia-20-22|issn=1331-4343|doi-access=free}}</ref><blockquote><math>2r \le D \le 2R</math>

<math>R \le \frac{\sqrt{3}}{3} D</math>

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:<math>\text{Conv}\left ( \sum_n S_n \right ) = \sum_n \text{Conv} \left (S_n \right).</math>

In mathematical terminology, the [[operation (mathematics)|operation]]s of Minkowski summation and of forming [[convex hull]]s are [[commutativity|commuting]] operations.<ref>Theorem&nbsp;3 (pages&nbsp;562–563): {{cite news|first1=M.|last1=Krein|authorlink1author-link1=Mark Krein|first2=V.|last2=Šmulian|year=1940|title=On regularly convex sets in the space conjugate to a Banach space|journal=Annals of Mathematics |series=Second Series| volume=41 |pages=556–583|jstor=1968735|doi=10.2307/1968735}}</ref><ref name="Schneider">For the commutativity of [[Minkowski addition]] and [[convex hull|convexification]], see Theorem&nbsp;1.1.2 (pages&nbsp;2–3) in Schneider; this reference discusses much of the literature on the [[convex hull]]s of [[Minkowski addition|Minkowski]] [[sumset]]s in its "Chapter&nbsp;3 Minkowski addition" (pages&nbsp;126–196): {{cite book|last=Schneider|first=Rolf|title=Convex bodies: The Brunn–Minkowski theory|series=Encyclopedia of mathematics and its applications|volume=44|publisher=Cambridge&nbsp;University Press|___location=Cambridge|year=1993|pages=xiv+490|isbn=0-521-35220-7|mr=1216521|url=https://archive.org/details/convexbodiesbrun0000schn}}</ref>

=== Minkowski sums of convex sets ===