Minuscule weights of parabolic sub-root systems are not far from dominant 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.
 A: In the latest version of our paper "A positive formula for the Ehrhart-like polynomials from root system chip-firing," with Alex Postnikov (https://arxiv.org/abs/1803.08472) we explicitly discuss some of these questions.
Let $\mathcal{P}_{\Phi^\vee}:=\mathrm{ConvexHull}(\Phi^\vee)$ be the root polytope of the dual root system $\Phi^\vee$ (this is different notation than in the original question).
We show the following:
Lemma 1: Let $\{0\}\neq U \subseteq V$ be a nonzero subspace of $V$ spanned by a subset of $\Phi^\vee$. Set $\Phi^\vee_U := \Phi^\vee\cap U$, a sub-root system of $\Phi^\vee$. Let $\pi_U\colon V\to U$ be the orthogonal projection. Then there exists a constant $1\leq \kappa < 2$ such that $\pi_U(\mathcal{P}_{\Phi^\vee})\subseteq \kappa\cdot \mathcal{P}_{\Phi^\vee_U}$.
Note that the polar dual of $\mathcal{P}_{\Phi^\vee}$ is $\mathcal{P}^{*}_{\Phi^\vee} = \{v\in V\colon \langle v,\alpha^\vee\rangle \leq 1 \textrm{ for all $\alpha \in \Phi$}\}$. Taking polar duals, Lemma 1 then says that there is some $1\leq \kappa < 2$ such that $\mathcal{P}^*_{\Phi^\vee_U}\subseteq \kappa\cdot \mathcal{P}^*_{\Phi^\vee}$. The minuscule weights of $\Phi_U$ are (a subset of) the dominant vertices of $\mathcal{P}^*_{\Phi^\vee_U}$: they are just the nonzero $u\in U$ which have $\langle u,\alpha^\vee\rangle \in\{0,1\}$ for all $\alpha \in \Phi^+$. So Lemma 1 indeed implies, as is conjectured in the original question, that $\langle u, \alpha^\vee\rangle > -2$ for any positive root $\alpha\in \Phi^+$ and $u \in U$ zero or a minuscule weight of $\Phi_U$. (Actually the original question only considered simple roots $\alpha_j$.)
Our interest in Lemma 1 is its relation to a lemma concerning an integrality property of slices of permutohedron, which I will now explain.
Let $P:=\{v \in V\colon \langle v, \alpha^\vee\rangle \in \mathbb{Z} \textrm{ for all $\alpha\in\Phi$}\}$ the weight lattice of $\Phi$ and $Q:=\mathrm{Span}_{\mathbb{Z}}$ the root lattice of $\Phi$. Let $W$ be the Weyl group of $\Phi$. For $\lambda \in P$, let $\Pi(\lambda) := \mathrm{ConvexHull}W(\lambda)$ denote the permutohedron of $\lambda$. We use $\Pi^Q(\lambda):=\Pi(\lambda)\cap(Q+\lambda)$ for the root lattice points in $\Pi(\lambda)$. 
The integrality lemma in question is:
Lemma 2: Let $\lambda \in P$, $\mu \in Q+\lambda$, and $X\subseteq \Phi$. Suppose that $\Pi(\lambda)\cap (\mu+\mathrm{Span}_{\mathbb{R}}
(X))\neq \varnothing$. Then $\Pi^Q(\lambda)\cap(\mu+\mathrm{Span}_{\mathbb{R}}(X))\neq \varnothing$.
In the above-linked paper, Alex and I show in a uniform way that Lemma 1 implies Lemma 2. The significance of Lemma 2 is that it leads to a formula for $\#\Pi^Q(\lambda+k\rho)$ where $\rho=\frac{1}{2}\sum_{\alpha\in\Phi^+}\alpha$ is the Weyl vector and $\lambda$ is a dominant weight.
However, none of this work actually addresses the main thrust of the original question, because unfortunately our proof of Lemma 1 is not uniform: we just check, in the appendix of that paper, that for all the (irreducible) root systems and all the (maximal parabolic) subspaces, the relevant containment of projections of root polytopes holds for some $\kappa < 2$.
I am still very much interested in ideas about a uniform proof of Lemma 1.
