0
$\begingroup$

This is related to another question of mine

Let $V=\Bbb R^n$. Morelli defined the (commutative unital) ring $L(V)$ to be the additive group generated by the indicator functions of convex polytopes in $V$ with multiplication induced by Minkowski sum.

Question: Are there any positive-dimensional polytopes $P$ satisfy the following condition (*)? If there are, is there a “good” (take that to mean what you will) characterization of them?

(*) Viewed as elements of $L(V)$, the set of faces of $P$ (including $P$ itself) is algebraically independent (over $\Bbb Z$).

Example: (*) is always false for zonotopes essentially by definition.

$\endgroup$
  • $\begingroup$ what is the addition operation in the ring $L(V)$? $\endgroup$ – Yossi Lonke Nov 21 '18 at 8:05
  • $\begingroup$ @YossiLonke Minkowski sum $\endgroup$ – Avi Steiner Nov 21 '18 at 13:46
  • $\begingroup$ You write that "multiplication is induced by Minkowski sum." In a ring, there are two operations: multiplication and addition. So I repeat my question: what is the addition operation in this ring? $\endgroup$ – Yossi Lonke Nov 21 '18 at 22:06
  • $\begingroup$ @YossiLonke The multiplication operation is minkowski sum, while the addition operation is function addition $\endgroup$ – Avi Steiner Nov 22 '18 at 2:01

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.