---
tag: SXS6D87PHE
kind: concept
name: Minkowski addition
msc: [52A20, 52A05]
sources:
  - url: https://en.wikipedia.org/wiki/Minkowski_addition
same_as:
  wikidata: Q1322294
  wikipedia: Minkowski addition
---

## Summary

The Minkowski sum of two subsets $A$ and $B$ of a vector space is the set $A + B$ of all sums $a + b$ with $a \in A$ and $b \in B$.
Together with scaling $\lambda A = \set{\lambda a : a \in A}$, it makes the convex bodies of $\R^n$ into a cone-like algebraic structure.
Volume behaves well under this addition, which leads to the Brunn–Minkowski inequality and the theory of mixed volumes.

## Explanations

### L1 human

To add two shapes, place a copy of the second shape at every point of the first and take everything covered.
Adding a disk of radius $r$ to a square gives a square with rounded corners, thickened by $r$ on every side.
Adding two segments in different directions gives a parallelogram.
Adding a shape to a single point just moves the shape.