Reformulation of inequality constraint Version 1
Let $\pi$ be a given permutation of the integers $\{1,\ldots,n\}$ and let 
$$\mathcal{X}=\{x\in\mathbb{R}_{+}^{n} \mid x_{\pi(1)}\geq\cdots\geq x_{\pi(n)}\}$$
Suppose we seek some $a \in \mathbb{R}$ and $b \in \mathbb{R}^{n}$, that
\begin{align*}
a+b^\top x<0,\quad \forall x \in \mathcal{X}.
\end{align*}
Can this be reformulated in sufficient and necessary conditions (without the set $\mathcal{X}$)?

Version 2
Let $\pi$ be a given permutation of the integers $\{1,\ldots,n\}$.
Suppose we seek some $a \in \mathbb{R}$, $b \in \mathbb{R}^{n}$, and $d>0$ such that
\begin{align*}
a+b^\top x<0,\quad \forall x \in \mathcal{X},
\end{align*}
where 
$$\mathcal{X}=\{x\in\mathbb{R}_{+}^{n} \mid x_{\pi(1)}\geq\cdots\geq x_{\pi(n)}, \mathbb{1}^\top x \geq d\}.$$
Can this be reformulated in sufficient and necessary conditions (without the set $\mathcal{X}$)?

Comment: Any non trivial sufficient condition are also welcome. Ideally, it would be nice if existence of such $a, b$ (and $d$), could be formulated as the feasibility of a convex program.
 A: Given the $(n-1) \times n$ oriented incidence matrix
$$\mathrm C := \begin{bmatrix} -1 & 1 & 0 & 0 & \dots & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & \dots & 0 & 0 & 0\\ 0 & 0 & -1 & 1 & \dots & 0 & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & \dots & -1 & 1 & 0\\ 0 & 0 & 0 & 0 & \dots & 0 & -1 & 1\end{bmatrix}$$
and an $n \times n$ permutation matrix $\rm P$, we would like to find a vector $\mathrm v \in \mathbb R^n \setminus \{0_n\}$ for which the following linear program (LP) in $\mathrm x \in \mathbb R^n$
$$\begin{array}{ll} \text{maximize} & \mathrm v^\top \mathrm x\\ \text{subject to} & \mathrm C \mathrm P \mathrm x \leq 0_{n-1}\\ & \mathrm x \geq 0_n\end{array}$$
is bounded. The dual of the LP above is the following feasibility LP in $\mathrm y \in \mathbb R^{n-1}$
$$\begin{array}{ll} \text{minimize} & \mathrm 0\\ \text{subject to} & ( \mathrm C \mathrm P )^\top \mathrm y \geq \mathrm v\\ & \mathrm y \geq 0_{n-1}\end{array}$$
Let $\mathrm y = 1_{n-1}$ and $\mathrm v = \color{blue}{( \mathrm C \mathrm P )^\top 1_{n-1}}$. The dual LP's constraints are then satisfied. Since the dual is feasible, the primal is bounded. The value of the maximum (of the primal) is $0$.
