15
$\begingroup$

I was wondering if such a concept was used anywhere. What I was thinking of is this. Consider two vectors spaces $V,W$ and convex sets $C_1 \subseteq V$ and $C_2 \subseteq W$ if we define $C_1 \otimes C_2 := \text{Convex Hull}(\{c_1 \otimes c_2 \in C_1 \otimes C_2 : c_1 \in C_1,c_2 \in C_2 \}) $. If $C_1$ and $C_2$ are bounded convex sets then so is $C_1 \otimes C_2$. Also, I believe that it is the case that if $a$ is a vertex of $C_1$ and $b$ is a vertex of $C_2$ then $a \otimes b$ is a vertex of $C_1 \otimes C_2$ and all vertices of $C_1 \otimes C_2$ can be constructed in this way.

Does anybody know of a construction like this actually being used anywhere? In particularly interested in the case where $C_1$ is given by as the solution to a list of inequalities $\alpha_i(v) \leq 1$, $\alpha_i \in V^*$, and $C_2$ is likewise given by a list $\beta_j(w) \leq 1$, $\beta_j \in W^*$. Then I can see that $C_1 \otimes C_2$ is contained in the convex set given by the inequalities $\alpha_i \otimes \beta_j \leq 1$. I've been try and failing at showing that $C_1 \otimes C_2$ is in fact exactly equal to the solution of these inequalities. Any advice that may be helpful would be appreciated.

$\endgroup$
3
  • 2
    $\begingroup$ Possibly related: mathoverflow.net/questions/115677 $\endgroup$ Mar 30, 2016 at 0:44
  • $\begingroup$ In the definition of $C_1\otimes C_2$ you mean to write $c_1\otimes c_2\in C_1\times C_2$? Also, $c_1\otimes c_2$ is the same as $(c_1,c_2)$, right? $\endgroup$
    – Dirk
    Feb 2, 2018 at 9:18
  • $\begingroup$ No, I meant the tensor product of the vectors c1 and c2 $\endgroup$
    – Onye
    Feb 3, 2018 at 9:47

1 Answer 1

5
$\begingroup$

This is well-documented in the case where $C_1$, $C_2$ are unit balls for some norm, and in that case your $C_1 \otimes C_2$ is the unit ball for the so-called projective norm. The set you compare to in the last paragraph is the unit ball for the so-called injective norm, which indeed is a different norm. Googling these keywords should give many results.

Also, you may look at Chapter 4.1.4 in the book "G. Aubrun & S. Szarek, Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory", where we consider such projective tensor products for general convex sets.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.