Triangle Inequalities

This problem came about because someone mentioned they thought an inequality of the form $P^3 \ge k A$ was true. What's wrong with this at a quick glance? The units don't work out! It doesn't make sense that you could relate $\text{units}^3$ and $\text{units}^2$ in this way; that's the intuitive reason why $n=2$ gives the only valid inequality here.

For a more algebraic approach, note that $P^2 = 12\sqrt{3}\cdot A$ holds true for an equilateral triangle, so we can have $\frac{P^n}{A} = \left(12\sqrt{3}\right)^{n/2} \cdot A^{n/2 - 1}.$ Taking $A \rightarrow 0$ and $A\rightarrow \infty$ shows it is impossible for this ratio to always be $\ge k.$ (It's probably worth handling the $n=1$ case separately, but it also follows from the same logic.)