Let $\Phi$ be a crystallographic root system in an $n$-dimensional Euclidean vector space $(V,\langle\cdot,\cdot\rangle)$. For a root $\alpha\in \Phi$ we use $\alpha^\vee := \frac{2}{\langle \alpha,\alpha\rangle}\alpha$ to denote the corresponding co-root. Suppose that the simple roots of $\Phi$ are $\alpha_1,\ldots,\alpha_n$. For $I\subseteq[n]$, let $\Phi_I$ denote the corresponding parabolic sub-root system, i.e., $\Phi_I := \Phi\cap \mathrm{Span}_{\mathbb{R}}\{\alpha_i\colon i\in I\}$. Use $\Phi^+_I$ for the positive roots of $\Phi_I$ (with respect to the choice of simple roots $\{\alpha_i\colon i\in I\}$).

**Claim**: Let $I\subseteq [n]$. Let $u = \sum_{i\in I}a_i\alpha_i$ be an element of $\mathrm{Span}_{\mathbb{R}}\{\alpha_i\colon i\in I\}$ which is zero or a *minuscule weight* of $\Phi_I$, i.e., which satisfies $\langle u,\alpha^\vee\rangle \in\{0,1\}$ for all $\alpha\in \Phi^+_I$. (Note crucially that $u$ need not be an integral weight of $\Phi$.) Then $\langle u,\alpha_j^\vee\rangle > -2$ for all $j\in[n]$.

I know that the above claim is true thanks to a careful analysis of the classical types together with exhaustive computation for the exceptional types.

**Question 1**: Is there a conceptual, uniform proof of the above claim?

Now define the quantity $$M_{\Phi} := -\mathrm{min}\{\langle u,\alpha_j^\vee\rangle\colon I\subseteq [n], u=\sum_{i\in I}a_i\alpha_i \textrm{ is zero or a minuscule weight of $\Phi_I$},j\in[n]\}.$$

For example, here is the value of $M_\Phi$ for all irreducible root systems, together with the minimizing $I$ and $j$, following Bourbaki's numbering of the nodes of the Dynkin diagram:

- $\Phi = A_{2m+1}$: $M_{\Phi}= \frac{2m}{m+1}$ with $I=\{1,\ldots,m,m+2,\ldots,2m+1\}$ and $j=m+1$;
- $\Phi = A_{2m}$: $M_{\Phi}= \frac{m}{m+1}+\frac{m-1}{m}$ with $I=\{1,\ldots,m,m+2,\ldots,2m\}$ and $j=m+1$ (or also the symmetric $I=\{1,\ldots,m-1,m+1,\ldots,2m\}$ and $j=m$);
- $\Phi = B_n$: $M_{\Phi}= \frac{2n-2}{n}$ with $I=\{1,\ldots,n-1\}$ and $j=n$;
- $\Phi = C_n$: $M_{\Phi}= \frac{2n-3}{n-1}$ with $I=\{1,\ldots,n-2,n\}$ and $j=n-1$;
- $\Phi = D_n$: $M_{\Phi}= \frac{2n-5}{n-2}$ with $I=\{1,\ldots,n-3,n-1,n\}$ and $j=n-2$;
- $\Phi = G_2$: $M_{\Phi}= \frac{3}{2}$ with $I=\{2\}$ and $j=1$;
- $\Phi = F_4$: $M_{\Phi}= \frac{11}{6}$ with $I=\{1,2,4\}$ and $j=3$;
- $\Phi = E_6$: $M_{\Phi}= \frac{11}{6}$ with $I=\{1,2,3,5,6\}$ and $j=4$;
- $\Phi = E_7$: $M_{\Phi}= \frac{23}{12}$ with $I=\{1,2,3,5,6,7\}$ and $j=4$;
- $\Phi = E_8$: $M_{\Phi}= \frac{59}{30}$ with $I=\{1,2,3,5,6,7,8\}$ and $j=4$.

**Question 2**: Is there a uniform, root-theoretic formula for $M_{\Phi}$?

EDIT 1:

For context on the significance of this question, please see https://arxiv.org/abs/1803.08472, especially Remark 2.11. The inequality in the above claim is closely related to a certain integrality property of slices of $W$-permutohedra.

EDIT 2:

In some discussions with Alex Postnikov, we found a way of reformulating this question (in terms of alcoved polytopes).

For a root system $\Phi$, define the following polytope:
$$\mathcal{P}_{\Phi}:=\{v \in \mathrm{Span}_{\mathbb{R}}(\Phi)\colon |\langle v,\alpha^\vee\rangle | \leq 1 \textrm{ for all $\alpha\in\Phi$}\}.$$
Note that $\mathcal{P}_{\Phi}$ is not a permutohedron; if $\Phi$ is irreducible, it is an *alcoved polytope* in the sense of Lam-Postnikov (https://arxiv.org/abs/1202.4015).

**Conjecture**: Let $\Phi$ be a (crystallographic) root system in a vector space $V$. Let $U\subseteq V$ be a subspace of $V$ spanned by some subset of roots. Set $\Phi' := \Phi\cap U$, a sub-root system. Then:
$$\mathcal{P}_{\Phi'}\subseteq \mathrm{interior}(2\cdot\mathcal{P}_{\Phi}).$$

It is easy to see how this conjecture implies the above claim. I am pretty sure an exhaustive analysis could also prove this conjecture. But the question I really have is:

**Question 3**: Is there a uniform proof of the above conjecture?

EDIT 3:

Here are some updates about question 3. Note that to prove the conjecture it suffices to restrict to the case where $\Phi'$ is a maximal parabolic sub-root system of $\Phi$.

The conjecture amounts to the assertion that the minimal $\kappa$ for which $\mathcal{P}_{\Phi'}\subseteq \kappa \cdot \mathcal{P}_{\Phi}$ is strictly less than $2$. Hence rather than study $M_{\Phi}$ as I defined it above, it might make more sense to try to give a root-theoretic formula for this minimal $\kappa$ in terms of a choice of $\Phi$ and a maximal parabolic of $\Phi$ (i.e., a node of the Dynkin diagram).

Here are some pictures of how these $\mathcal{P}_{\Phi}$ and $\mathcal{P}_{\Phi'}$ look.

For $\Phi=B_3$ and $\Phi'$ the maximal parabolic corresponding to node $3$, we show $\mathcal{P}_{\Phi}$ in blue and we show $\mathcal{P}_{\Phi'}$ in red:

For $\Phi=C_3$ and $\Phi'$ the maximal parabolic corresponding to node $2$, we show $\mathcal{P}_{\Phi}$ in blue and we show $\mathcal{P}_{\Phi'}$ in red:

Of course, it suffices to restrict $\mathcal{P}_{\Phi}$ to $\mathrm{Span}_{\mathbb{R}}(\Phi')$; here is what that looks like for the previous two examples, where we also depict $2\cdot\mathcal{P}_{\Phi}$ in green to verify the requisite containment:

By restricting like this we can also depict rank 4 examples. For example, here is $\Phi=A_4$ for the parabolic corresponding to node 2 (or node 3):

And here is $\Phi=D_4$ for the parabolic corresponding to the trivalent node 2:

This is a cuboctahedron inscribed in a cube. Note that the polytope $\mathcal{P}_{\Phi}$ is the polar dual of the ``root polytope'' $\mathrm{ConvHull}(\Phi^\vee)$. Here is the polar dual of the previous example:

This is an octahedron inscribed in a rhombic dodecahedron.