structural:reformatting
Convex hulls of Minkowski sums

Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition:

Let S, S be subsets of a real vector-space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls Conv (S 1 + S 2) = Conv (S 1) + Conv (S 2). {\displaystyle \operatorname {Conv} (S_{1}+S_{2})=\operatorname {Conv} (S_{1})+\operatorname {Conv} (S_{2}).}

This result holds more generally for each finite collection of non-empty sets: Conv (∑ n S n) = ∑ n Conv (S n). {\displaystyle {\text{Conv}}\left(\sum _{n}S_{n}\right)=\sum _{n}{\text{Conv}}\left(S_{n}\right).}

In mathematical terminology, the operations of Minkowski summation and of forming convex hulls are commuting operations.