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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