A set of the form $$\left\{x \in \Bbb R^n\mid \alpha \leq a^T x \leq \beta \right\}$$ is called a slab. A slab is an intersection of two halfspaces. Hence, it is a convex set.
Can someone explain why a slab is an intersection of two halfspaces? I can't understand it.
A slab is a set of the form $$S = \{x\in\mathbb R^n : \alpha\le a^{T} x\le\beta\}$$ for some fixed $a = (a_1,a_2,\ldots,a_n)\in\mathbb R^n$ and $\alpha,\beta\in\mathbb R$.
$S$ is the intersection of $$H_1 = \{x\in\mathbb R^n : a^{T} x\ge\alpha\}$$ and $$H_2 = \{x\in\mathbb R^n : a^{T} x\le\beta\}$$ i.e. $S = H_1\cap H_2$. $H_1$ and $H_2$ are halfspaces. Hence, $S$ is convex.
