# would relation induced subset break cartesian product - homomorphism of convex hull?

Mathematics Asked by peng yu on December 18, 2020

Given two set of vectors $$A, B subset mathbb{R}^n$$, and a relation $$D subseteq A times B$$, here $$times$$ denote cartesian product.

$$CH(Atimes B) = CH(A) times CH(B)$$, $$CH$$ is the convex hull operation. for which we can think $$CH$$ is a homomorphism of $$times$$.

Any subset $$A’ subseteq A$$ would induce a subset $$B’$$ via relation $$D$$: $${b|(a, b) in D, a in A’}$$.

Also since $$A, B in mathbb{R}^n$$, we can merge a set into a vector $$S(A) = sum_{a in A} a$$, and we define $$S(emptyset) = vec{0}$$.

$$E = cup_{A’ subseteq A} { S(A’) } times {S(B”) | B” subseteq B’}$$, here $$B’$$ is the subset induced from $$A’$$ using relation $$D$$.

Let $$CP(E)$$ denote the subset points of $$E$$ in convex position.

Would $$CP({S(A’)| A’ subseteq A})$$ the same as $${a|(a,cdot) in CP(E)})$$ ?

The reason I’m interested in this is I’m trying to "generalize" the homomorphism to any relation induced.

If the relation $$D = A times B$$.

Then $$E = cup_{A’ subseteq A} {S(A’)} times {S(B’)|B’subseteq B}$$, the conclusion would be true.

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