What does the term "geometry" means in a Banach space？
I heard someone saying that it's popular to study geometry on Banach space in the past several years. He showed me some important results. However, I am still not clear about what it means by geometry in functional analysis. Is convexity a geometrical property? In what occasion can I say that a theorem is about a geometric property?
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$\begingroup$Yeah, convexity is a geometric property.
It often means you study properties of the space by looking at geometric properties of the unit ball or other convex sets. As an important example, there's the notion of dentability. Fix a Banach Space $X$ and closed convex subset $C$.
Let $f \in X^* \setminus \lbrace 0 \rbrace$, and suppose $\alpha = \sup f(C) < \infty$. If we define $S$ by $$S = f^{-1}[\beta, \alpha] \cap C$$ for some $\beta < \alpha$, then $S$ is a slice. We say $C$ is dentable if it admits slices of arbitrarily small diameter. We also say $X$ is dentable if every closed bounded non-empty convex subset is dentable.
Dentability is considered a geometric property. Slices are very straightforward, in the sense that you're basically just "slicing" off part of the convex set with a hyperplane defined by a functional. Dentability means you want to be able to make it arbitrarily small in diameter, which you can visualise by enveloping in balls of arbitrarily small diameter.
Dentability of the space, as it turns out, is equivalent to the Radon-Nikodym property, a property of vector measures, holding on the Banach Space. So, a geometric property implies a measure theoretic property. That's an example of a geometry of Banach Spaces result.
A weaker form of dentability, weak$^*$ dentability, is also used to characterise the Mazur Intersection Property (which a space has, if every closed bounded convex subset can be realised as the intersection of balls), as well as define Asplund spaces, on which convex functions have highly desirable differentiability properties.
As another example, the Bishop-Phelps theorem, guaranteeing density of support points on closed convex sets is also typically proven geometrically. It's done by intersecting pointed cones with convex sets, in a nested way, so that you obtain a nested family of (weakly) closed sets with null-convergent diameter. Completeness implies a (unique) point of intersection, at which the cone touches the convex set at one point. Then separation theorem applied to the set and the cone yields a supporting hyperplane (separation theorem is, in a sense, the most fundamental result for geometry of Banach Spaces).
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