A remark on Edgar's extremal integral representation theorem
Factoring the identity operator on a subspace of
Weak Distances between Random Subproportional Quotients of
Lower estimates for weak distances between finite-dimensional Banach spaces of the same dimension are investigated. It is proved that the weak distance between a random pair of n-dimensional quotients of is greater than or equal to c√(n/log³n).
Volumetric invariants and operators on random families of Banach spaces
The geometry of random projections of centrally symmetric convex bodies in is studied. It is shown that if for such a body K the Euclidean ball is the ellipsoid of minimal volume containing it and a random n-dimensional projection is “far” from then the (random) body B is as “rigid” as its “distance” to permits. The result holds for the full range of dimensions 1 ≤ n ≤ λN, for arbitrary λ ∈ (0,1).
On the geometry of proportional quotients of
We compare various constructions of random proportional quotients of (i.e., with the dimension of the quotient roughly equal to a fixed proportion of m as m → ∞) and show that several of those constructions are equivalent. As a consequence of our approach we conclude that the most natural “geometric” models possess a number of asymptotically extremal properties, some of which were hitherto not known for any model.
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