# Law of cosines in normed spaces which are not inner product spaces?

I have a finite dimensional normed vector space $$V$$ over $$\mathbb{R}$$. In practice I care mainly about the $$p$$-norm for $$p\in[1,\infty]$$, but there is no need to specialize to this case yet.

I'm interested in getting lower bounds on $$\lVert a + b\rVert - \lVert a\rVert$$. I'm also fine with weaker lower bounds on $$\lVert a + b\rVert^k - \lVert a\rVert^k$$ if this makes things easier. In the $$\ell_2$$ norm you can relate these two quantities with the polarization identity/law of cosines. You in particular have:

$$\lVert a + b\rVert^2 - \lVert a\rVert^2 = \lVert b\rVert^2 +2\langle a, b\rangle$$

We cannot hope for an identity along these lines, as it well known that the polarization identity implies that the norm comes from an inner product (and therefore for $$\ell_p$$ norms is $$\ell_2$$). I would be happy enough with a lower bound of the form $$\lVert b\rVert^k + f(a, b, k)$$ though, provided that the bound is relatively tight (and $$f(k, a, b)$$ is relatively simple). Does such a bound exist?