**Question 1.** Can a reflexive Delzant polytope of some dimension contain a $2$-face with more than $11$ edges?


**Motivation.** I would like more generally to get an answer to the following question:


**Question 2.** Suppose $X$ is a smooth Fano variety with a $\mathbb C^\*$-action. Let $Y\subset X$ be the connected component of $X^{\mathbb C^\*}$ on which all the weights of the action are positive. Is it true that variety $Y$ can be deformed to a Fano variety?


A positive answer to the first question gives a negative to the second one.


**Explanation.** Indeed, a reflexive Delzant polytope by definition corresponds to a toric Fano. For any face of a Delzant polytope we can find a rational linear support function. This will give us a $\mathbb C^\*$ action as in Question 2. A surface that deforms to a Fano surface has $b\_2\le 9$. Finally, a toric surface corresponding to a polygon with $n$ sides has $b\_2=n-2$.



2-faces of reflexive Delzant polytopes
Haase and Melnikov have proved [here](https://arxiv.org/abs/math/0406485) that every lattice polytope can be realized as a face of some reflexive polytope. They do this by an iterative procedure that increases the dimension by one until the polytope becomes reflexive. At each step the new polytope is a wedge over a facet of the previous one, and one can show that this operation takes Delzant polytopes to Delzant polytopes. So if you start with a smooth polygon with more than 11 edges will produce a positive answer to your first question.


