Minkowski sum

Definition Suppose A and B are sets in a vector spaceMathworldPlanetmath V over a field K, and suppose λK. Then

A+B = {a+baA,bB},
A-B = {a-baA,bB},
λA = {λaaA},
-A = (-1)A.

The set A+B is called the Minkowski sum of A and B. If either A or B is a single point (a singleton), say B={x}, then we write A+x instead of A+{x}. Similarly we define A-x, x-A and x+A.


Suppose A,B, V, and λ are as above. Then

  • A+B=B+A

  • λ(A+B)=λA+λB

  • 2AA+A, 3AA+A+A, etc, but in general, A+A2A. (Consider A={(0,0),(0,1)} in 2.)

Title Minkowski sum
Canonical name MinkowskiSum
Date of creation 2013-03-22 15:16:22
Last modified on 2013-03-22 15:16:22
Owner matte (1858)
Last modified by matte (1858)
Numerical id 4
Author matte (1858)
Entry type Definition
Classification msc 20-00
Classification msc 15-00
Classification msc 13-00
Classification msc 16-00
Related topic VectorSpace
Related topic Sumset