Convex Hulls of Minkowski Sums
Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition:
- For all subsets S1 and S2 of a real vector-space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls
- Conv(S1 + S2) = Conv(S1) + Conv(S2).
This result holds more generally for each finite collection of non-empty sets
- Conv(∑Sn) = ∑Conv(Sn).
In mathematical terminology, the operations of Minkowski summation and of forming convex hulls are commuting operations.
If S is a convex set then also is a convex set; furthermore
- for every .
Conversely, if this "distributive property" holds for all non-negative real numbers, then the set is convex. The figure shows an example of an non-convex set for which A + A ≠ 2A.
Minkowski sums act linearly on the perimeter of two-dimensional convex bodies: the perimeter of the sum equals the sum of perimeters. Additionally, if K is (the interior of) a curve of constant width, then the Minkowski sum of K 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.
Read more about this topic: Minkowski Addition
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