Hello all, I'm trying to find a good resource for a discussion on the relation between say, the p-norm of a vector (from a finite dimensional vector space) and its Euclidean norm. In my search on the internet and in various books, I only encounter basic, standard inequalities such as the Cauchy-Schwarz and Holder's inequality.

Are there textbooks that go more in depth than these two?

In particular, I'm interested in the following: if I have two unit vectors $\psi$ and $\phi$ (from $R^d$, say), that are $\epsilon$-close, meaning that $\|\psi - \phi\|_2 \leq \epsilon$, then what can one say about $\|\psi\|_p - \|\phi\|_p$? Intuitively, they must be close as well, but does the closeness depend on $d$, the dimension of the vector-space?

Any references or links or pointers would be greatly appreciated!

Thanks, Henry