When is a halfspace a subset of another halfspace? Let's say we have the following halfspaces:
$$H_1=\{x\mid a^Tx\leq b\}$$ and $$H_2=\{x\mid\tilde{a}^Tx\leq \tilde{b}\}$$

I want to find the conditions that need to hold such that $H_1\subseteq H_2$. So these halfspaces are defined by the hyperplanes, say, 
$$A_1=\{x\mid a^Tx=b\}$$ and
$$A_2=\{x\mid\tilde{a}^T x=\tilde{b}\}$$
Intuitively I understand that $H_1\subseteq H_2$ demands the hyperplanes to be parallel or equivalently the normal vectors $a$ and $\tilde{a}$ to be parallel, so:
\begin{eqnarray*}
&amp;&amp;A_1\parallel A_2
\\&amp;\Leftrightarrow&amp; a\parallel \tilde{a}
\\&amp;\Leftrightarrow&amp; \exists t\in \Re\text{ such that }\tilde{a}=ta
\end{eqnarray*}
The hypothesis $H_1\subseteq H_2$ also requires the $H_2$ to lie above $H_1$ which restricts us to consider $\tilde{b}\ge tb$ for positive values of $t$ ($t&gt;0$). But how can I formally prove that these two conditions are enough and necessary?
For the general case, to appear $t&gt; 0$, let us normalize the vectors $a, b$. Then we have,
$H_1=\{x\mid a^Tx \le b\} = \{x\mid\frac{a^T}{\|a\|} \le \frac{b}{\|a\|}\}$.
Do the same for $H_2$ and do the same as above. We will show $t =\frac{\|a\|}{\|a'\|}$.
For the general case, to appear $t&gt; 0$, let us normalize the vectors $a, b$. Then we have,
$H_1=\{x\mid a^Tx \le b\} = \{x\mid\frac{a^T}{\|a\|} \le \frac{b}{\|a\|}\}$.
Do the same for $H_2$ and do the same as above. We will show $t =\frac{\|a\|}{\|a'\|}$.