Editing Minkowski addition (section)

== Convex hulls of Minkowski sums ==
Minkowski addition behaves well with respect to the operation of taking [[convex hull]]s, as shown by the following proposition:

{{block indent| For all non-empty subsets <math display="inline">S_1</math> and <math display="inline">S_2</math> of a real vector space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls:

<math display="block">\operatorname{Conv}(S_1 + S_2) = \operatorname{Conv}(S_1) + \operatorname{Conv}(S_2).</math>}}

This result holds more generally for any finite collection of non-empty sets:

<math display="block">\operatorname{Conv}\left(\sum{S_n}\right) = \sum\operatorname{Conv}(S_n).</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 journal|first1=M.|last1=Krein|author-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|issue=3 |pages=556–583|doi=10.2307/1968735|mr=2009 | jstor = 1968735}}</ref><ref>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 [[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=978-0-521-35220-8|mr=1216521|url=https://archive.org/details/convexbodiesbrun0000schn}}</ref>

If <math display="inline">S</math> is a convex set then <math>\mu S + \lambda S</math> is also a convex set; furthermore

<math display="block">\mu S + \lambda S = (\mu + \lambda)S</math>

for every <math display="inline">\mu,\lambda \geq 0</math>. Conversely, if this "[[distributive property]]" holds for all non-negative real numbers, <math display="inline">\mu</math> and <math display="inline">\lambda</math>, then the set is convex.<ref>Chapter 1: {{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=978-0-521-35220-8|mr=1216521|url=https://archive.org/details/convexbodiesbrun0000schn}}</ref>

[[File:Minkowskisum.svg|thumb|An example of a non-convex set such that <math display="inline">A + A \neq 2 A.</math>]]
The figure to the right shows an example of a non-convex set for which <math display="inline">A + A \subsetneq 2 A.</math>

An example in one dimension is: <math display="inline">B = [1, 2] \cup [4, 5].</math> It can be easily calculated that <math display="inline">2 B = [2, 4] \cup [8, 10]</math> but <math display="inline">B + B = [2, 4] \cup [5, 7] \cup [8, 10],</math> hence again <math display="inline">B + B \subsetneq 2 B.</math>

Minkowski sums act linearly on the perimeter of two-dimensional convex bodies: the perimeter of the sum equals the sum of perimeters. Additionally, if <math display="inline">K</math> is (the interior of) a [[curve of constant width]], then the Minkowski sum of <math display="inline">K</math> and of its 180° rotation is a disk. These two facts can be combined to give a short proof of [[Barbier's theorem]] on the perimeter of curves of constant width.<ref>[http://www.cut-the-knot.org/ctk/Barbier.shtml The Theorem of Barbier (Java)] at [[cut-the-knot]].</ref>