Complex Analytic and Formal Solutions of Real Analytic Equations in Cn.
Analytic and Polynominal Homomorphisms of Analytic Rings.
On extreme points of regular convex sets
Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces
Semianalytic and subanalytic sets
Arc-analytic functions.
Local analytic invariants
Regularization of star bodies by random hyperplane cut off
We present a general result on regularization of an arbitrary convex body (and more generally a star body), which gives and extends global forms of a number of well known local facts, like the low M*-estimates, large Euclidean sections of finite volume-ratio spaces and others.
Relations among analytic functions. I
Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let be a morphism of real analytic spaces, and let be a homomorphism of coherent modules over the induced ring homomorphism . We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations , , are upper semi-continuous in the analytic Zariski topology of . We prove semicontinuity in many cases (e.g. in the algebraic category)....
Relations among analytic functions. II
This is a sequel to “Relations among analytic functions I”, , , fasc. 1, [pp. 187-239]. We reduce to semicontinuity of local invariants the problem of finding solutions to systems of equations involving division and composition by analytic functions. We prove semicontinuity in several general cases : in the algebraic category, for “regular” mappings, and for module homomorphisms over a finite mapping.
Essentially-Euclidean convex bodies
In this note we introduce a notion of essentially-Euclidean normed spaces (and convex bodies). Roughly speaking, an n-dimensional space is λ-essentially-Euclidean (with 0 < λ < 1) if it has a [λn]-dimensional subspace which has further proportional-dimensional Euclidean subspaces of any proportion. We consider a space X₁ = (ℝⁿ,||·||₁) with the property that if a space X₂ = (ℝⁿ,||·||₂) is "not too far" from X₁ then there exists a [λn]-dimensional subspace E⊂ ℝⁿ such that E₁ = (E,||·||₁) and...
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