The local theory of normed spaces is a topic in the intersection of
Functional Analysis, Geometry of Convex Sets and Probability.
It deals with the fine structure of finite dimensional normed spaces or equivalently with that of high dimensional symmetric convex sets, which are sets of points in high dimensional space with the property that for any two points in the set, the whole segment joining them is also in the set.
The local theory of normed spaces is also connected with several other areas going as far as the theory of Algorithms. These connections are mostly through the methods of proof developed in the local theory.
One of the major breakthroughs is this area, in which researchers
from the Weizmann Institute played a major role, mostly during the 80-s,
was the investigation of
high dimensional subspaces and quotients of classical normed spaces;
equivalently, of sections and projections of symmetric convex sets.
Other surprising properties of high dimensional convex sets
were also discovered.
The results are usually counter intuitive, showing that high dimensional spaces are very different from the three dimensional space (some of us think) we live in. The proofs are mostly probabilistic: Typically one shows that objects with a certain property exist by showing that "most" of the relevant objects have this property, usually without being able to point at even one explicit object possessing the desired property.