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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