Week 4

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This week, I explored the proof of Discrete Schwartz Lemma, specifically its use of Perron's Method. Where to show , we show they both lie in a set , which is closed under minimums and is convex. We then show that is the infimum of thus . I wanted to see if the "opposite set" would still work to prove the theorem, so I defined and showed that it is closed under maximums and convex. But an issue pops up where there may exist a sequence such that such that

, and the arguement would not work. The reason I am doing this, is because for the cases where circle packings are no longer tangent, the set of possible radii assignments may no longer be and won't be closed under minimums. But the set may still be closed under maximums, so I wanted to see if the "opposite" set would work.

Additionally, I am also working through Stephanson's book on circle packings, and proved the theorem that for any combinatorial closed disk, there exists a unique (up to Mobius transformations) maximal hyperbolic circle packing on with the same combinatorial structure.