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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?

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