By using a multiscale analysis, we establish quantitative versions of the Besicovitch projection theorem (almost every projection of a purely unrectifiable set in the plane of finite length has measure zero) and a standard companion result, namely that any planar set with at least two projections of measure zero is purely unrectifiable. We illustrate these results by providing an explicit (but ...

In [GM2], Goresky and MacPherson defined and constructed intersection complexes for topological pseudomanifolds. The complexes are defined in the derived category of sheaves of modules over a constant ring sheaf. Since analytic spaces are of this category, algebraic varieties defined over C have intersection complexes. The intersection complex of a given variety has a variation depending on a s...

We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann’s theorem to the complex case by proving that complex intersection bodies of symmetric complex convex bodies are also convex. Other results include stability in the complex Busemann-Petty problem for arbitrary measures an...

In 1870 Georg Cantor proved that a 2s periodic complex valued function of a real variable coincides with the values of at most one trigonometric series. We present his proof and then survey some of the many one dimensional generalizations and extensions of Cantor's theorem. We also survey the situation in higher dimensions, where a great deal less is known. 1. Cantor's uniqueness theorem. In 18...

The theorem presented and proved in this paper can be viewed as an extension or improvement of our recent Complete Intersection Theorem [1] and may be called the Complete Nontrivial-Intersection Theorem. It goes considerably beyond the well-known Hilton Milner Theorem [10]. We put the result into the proper perspective with a brief sketch of the key steps in its development, beginning with the ...

In this note we present new proofs of three basic theorems in calculus. Although these theorems are well-known, in each proof we obtain something which seems to be unknown. We start with the Heine-Cantor theorem about uniform continuity and obtain explicitly the optimal delta for the given epsilon. We then proceed with the Weierstrass extreme value theorem and present two proofs of it: the “env...