I am interested in knowing about "geometric" properties shared by equivalent norms on a Banach space.
Here I mean "geometric" as opposed to topological, and probably in particular with reference to the intrinsic geometry of the unit ball or sphere.
Apologies for the vagueness. A pointer towards a reference will do just fine.
I can give a somewhat vague example of what I mean: Where $\{e_n\}$ represents some orthonormal basis in $L^2[0,1]$, the norms defined by the closed convex hulls of $\{e_n\}$ and $\{\frac{1}{n} e_n \}$ are inequivalent. This in-equivalence is represented geometrically by the vanishing of the diameter of the latter unit ball "in the limit," though that is really nothing more than a rephrasing of the definition of the equivalence of norms.
I guess one could rephrase my question in the following way: what geometric difference between unit balls arise in infinite dimensional normed vector spaces that are not apparent (from the perspective of norm equivalence) in the finite case?
Are there factors other than the decay of the norm of an arbitrary sequence points on the sphere? Having written this out, I now realize that the condition of being in-equivalent is (analytically) precisely that of the existence of a sequence of elements that eventually break the inequality $(c || f||_1 \leq ||f||_2 \leq C ||f||_1$) regardless of the constants used.
Are there other useful to know norm-equivalence invariants?