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By Igor Pak
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Additional info for Lectures on Discrete and Polyhedral Geometry
Example text
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].