Convex Hull - Minkowski Addition and Convex Hulls

Minkowski Addition and Convex Hulls

See also: Minkowski addition and Shapley–Folkman lemma

The operation of taking convex hulls behaves well with respect to the Minkowski addition of sets.

  • In a real vector-space, the Minkowski sum of two (non-empty) sets S1 and S2 is defined to be the set S1 + S2 formed by the addition of vectors element-wise from the summand-sets
S1 + S2 = { x1 + x2 : x1 ∈ S1 and x2 ∈ S2 }.

More generally, the Minkowski sum of a finite family of (non-empty) sets Sn is the set formed by element-wise addition of vectors

∑ Sn = { ∑ xn : xn ∈ Sn }.
  • 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 other words, the operations of Minkowski summation and of forming convex hulls are commuting operations.

These results show that Minkowski addition differs from the union operation of set theory; indeed, the union of two convex sets need not be convex: The inclusion Conv(S) ∪ Conv(T) ⊆ Conv(S ∪ T) is generally strict. The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "join" operation is the convex hull of the union of two convex sets

Conv(S)∨Conv(T) = Conv( S ∪ T ) = Conv( Conv(S) ∪ Conv(T) ).

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