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By Giuseppe Zampieri

Cauchy-Riemann (CR) geometry is the examine of manifolds built with a approach of CR-type equations. in comparison to the early days while the aim of CR geometry was once to provide instruments for the research of the lifestyles and regularity of suggestions to the $\bar\partial$-Neumann challenge, it has swiftly received a lifetime of its personal and has turned a big subject in differential geometry and the learn of non-linear partial differential equations. an entire knowing of contemporary CR geometry calls for wisdom of assorted subject matters akin to real/complex differential and symplectic geometry, foliation conception, the geometric idea of PDE's, and microlocal research. these days, the topic of CR geometry is particularly wealthy in effects, and the quantity of fabric required to arrive competence is formidable to graduate scholars who desire to examine it. although, the current ebook doesn't objective at introducing all of the themes of present curiosity in CR geometry. as a substitute, an try is made to be pleasant to the amateur by means of relocating, in a reasonably secure approach, from the weather of the speculation of holomorphic services in numerous advanced variables to complex issues akin to extendability of CR features, analytic discs, their infinitesimal deformations, and their lifts to the cotangent house. the alternative of issues offers a very good stability among a primary publicity to CR geometry and topics representing present study. Even a pro mathematician who desires to give a contribution to the topic of CR research and geometry will locate the alternative of subject matters appealing

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Now, let y be a fixed point in the interior of an edge e in X. Consider a sequence of generic points v outside of X which converge to y. , the rhombi do not degenerate in the limit. Denote by X ′ a polygon symmetric to X with respect to y. We assume that X has no parallel edges. Then X ∩ X ′ consists of an interval on the edge e and finitely many points uk . One can choose points v to approach y in the direction that is neither orthogonal to intervals (y, uk ), nor to the edge e. When the points v 11Here we implicitly use the fact that one can parameterize each path from y to z so that the distance functions f1 and f2 are piecewise linear.

2. Jordan curve C and an inscribed square. implies the claim for self-intersecting closed polygons as well, since taking any simple cycle in it suffices. 1 generalizes to higher dimensions. The answer is yes, but the proof is more delicate. In three dimensions this is called the Kakutani theorem; we prove it in the next section. In fact, much of the next section is based on various modifications and generalizations of the Kakutani theorem. 2. Inscribing triangles is easy. , X = ∂A. We say that an equilateral triangle is inscribed into X if there exist three distinct points y1 , y2, y3 ∈ X such that |y1y2 | = |y1 y3 | = |y2y3 |.

The Boros–F¨ uredi theorem is proved in [BorF]. Our proof follows a recent paper [Bukh]. 9). 5) is usually attributed to L´evy (1934). It was pointed out in [Fle] that the result was first discovered by Amp`ere in 1806. The proof we present is due to Hopf [Hop1]. Our presentation follows [Lyu, §34]. 7 is due to Goldberg and West (1985), and was further generalized a number of times. 12). 6] for further results, 9This is called cobweb equipartition. 38 proofs and references. Our presentation is a variation on several known proofs and was partly influenced by [Tot].

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