In linear programming where we seek to minimize $c^Tx \to \text{min}_{x\in P}!$ with respect to some inequality constraints, why do we choose $P$ in the form
$Ax \leq b$, $x \geq 0$
as the standard form of a linear program? In the geometrical view $\{x: Ax \leq b\}$ seems much more natural, why don't we use this, or a type II-inequality constraint? (I am aware that we can transform the inequalities, but this does not always preserve the dimension of the problem, and gives an overhead to the proofs)