Editing Minkowski addition (section)

== Example ==
[[File:Minkowski-sumex4.svg|thumb|Minkowski sum {{nowrap|''A'' + ''B''}}]]
For example, if we have two sets ''A'' and ''B'', each consisting of three position vectors (informally, three points), representing the [[vertex (geometry)|vertices]] of two [[triangle]]s in <math display="inline">\mathbb{R}^2</math>, with coordinates

<math display="block">A = \{(1,0), (0,1), (0,-1)\}</math>

and

<math display="block">B = \{(0,0), (1,1), (1,-1)\}</math>

then their Minkowski sum is

<math display="block">A + B = \{(1,0), (2,1), (2,-1), (0,1), (1,2), (1,0), (0,-1), (1,0), (1,-2)\},</math>

which comprises the vertices of a hexagon and its center.

For Minkowski addition, the {{em|zero set}}, <math display="inline">\{ 0 \},</math> containing only the [[zero vector]], 0, is an [[identity element]]: for every subset ''S'' of a vector space,

<math display="block">S + \{0\} = S.</math>

The [[empty set]] is important in Minkowski addition, because the empty set annihilates every other subset: for every subset ''S'' of a vector space, its sum with the empty set is empty:

<math display="block">S + \emptyset = \emptyset.</math>

For another example, consider the Minkowski sums of open or closed balls in the field <math display="inline">\mathbb{K},</math> which is either the [[real number]]s <math display="inline">\R</math> or [[complex number]]s <math display="inline">\C</math>. If <math display="inline">B_r:= \{ s \in \mathbb{K}: |s| \leq r \}</math> is the closed ball of radius <math display="inline">r \in [0, \infty]</math> centered at <math display="inline">0</math> in <math display="inline">\mathbb{K}</math> then for any <math display="inline">r, s \in [0, \infty]</math>, <math display="inline">B_r + B_s = B_{r+s}</math> and also <math display="inline">c B_r = B_{|c|r}</math> will hold for any scalar <math display="inline">c \in \mathbb{K}</math> such that the product <math display="inline">|c|r</math> is defined (which happens when <math display="inline">c \neq 0</math> or <math display="inline">r \neq \infty</math>). If <math display="inline">r</math>, <math display="inline">s</math>, and <math display="inline">c</math> are all non-zero then the same equalities would still hold had <math display="inline">B_r</math> been defined to be the open ball, rather than the closed ball, centered at 0 (the non-zero assumption is needed because the open ball of radius 0 is the empty set). The Minkowski sum of a closed ball and an open ball is an open ball. More generally, the Minkowski sum of an [[Open set|open subset]] with {{em|any}} other set will be an open subset.

If <math display="inline">G = \{ (x, 1/x) : 0 \neq x \in \R \}</math> is the [[Graph of a function|graph]] of <math display="inline">f(x) = \frac{1}{x}</math> and if and <math display="inline">Y = \{ 0 \} \times \R</math> is the <math display="inline">y</math>-axis in <math display="inline">X = \R^2</math> then the Minkowski sum of these two [[Closed set|closed subsets]] of the plane is the [[open set]] <math display="inline">G + Y = \{ (x, y) \in \R^2 : x \neq 0 \} = \R^2 \setminus Y</math> consisting of everything other than the <math display="inline">y</math>-axis. This shows that the Minkowski sum of two [[closed set]]s is not necessarily a closed set. However, the Minkowski sum of two closed subsets will be a closed subset if at least one of these sets is also a [[Compact space|compact subset]].