Download Algebraic geometry 04 Linear algebraic groups, invariant by A.N. Parshin (editor), I.R. Shafarevich (editor), V.L. PDF

By A.N. Parshin (editor), I.R. Shafarevich (editor), V.L. Popov, T.A. Springer, E.B. Vinberg

Contributions on heavily similar topics: the idea of linear algebraic teams and invariant thought, via recognized specialists within the fields. The e-book could be very important as a reference and examine consultant to graduate scholars and researchers in arithmetic and theoretical physics.

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Denote by X ′ the clockwise rotation of X around z by an angle π/3. Clearly, area(X) = area(X ′ ). Thus for every z in the interior of an edge in X, polygons X and X ′ intersect at z and at least one other point v. Denote by u the counterclockwise rotation of v around z. Then (z, u, v) is the desired inscribed triangle. The same argument works when z is a vertex as in the proposition. Indeed, if z = xi and ∠ xi−1 xi xi+1 ≥ π/3, then the interiors of the corresponding regions A and A′ intersect.

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 |.

2 (Dyson). Let P ⊂ R3 be a convex polytope containing the origin: O ∈ P − ∂P . Then there exists a centrally symmetric square inscribed into P . Proof. Let us prove that the surface S = ∂P of the polytope P contains a centrally symmetric polygon Q. 1. Let P ′ be a reflection of P in O, and let S ′ = ∂P ′ . Since O ∈ P , then P ∩ P ′ = ∅, and thus either P ′ = P or S ∩ S ′ = ∅. Indeed, if S ∩ S ′ = ∅, then one of the polytopes would be contained in another and thus has a greater volume. Assume for now that the intersection W = P ∩ P ′ is one-dimensional, and moreover a union of non-intersecting polygons Q1 , .

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