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Today we talk about my favourite math theorem: the Ham Sandwich theorem. Consider a sandwich with three components. Then the theorem claims there exists some plane in three dimensions, where if I slice along that plane then I cut all three components exactly in half. We will see repeated use of the Intermediate Value Theorem in the simpler cases (which we call the pancake theorem), and then the more powerful Borsuk-Ulam theorem to prove the main result.
0:00 The Ham Sandwich Theorem
0:56 One Pancake and the Intermediate Value Theorem
2:12 The two pancakes theorem
3:40 Borsuk-Ulam Theorem
6:07 Proving the Ham Sandwich Theorem
10:45 Higher dimensions
11:24 Brilliant.org/TreforBazett
My previous video proving Sperner's Lemma and Brouwer's Fixed Point theorem (which can be shown are equivalent to Borsuk-Ulam): https://www.youtube.com/watch?v=oX9aPNF6_h8
I also have a previous video on Borsuk-Ulam and some other cool corollaries. It was actually my 10k subs video!: https://www.youtube.com/watch?v=z75B8iP1Y3g
Check out my MATH MERCH line in collaboration with Beautiful Equations
►https://beautifulequations.net/pages/trefor
COURSE PLAYLISTS:
►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS
►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6
►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m
► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n
►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd
►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa
►DIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw
►LAPLACE TRANSFORM: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1
►GAME THEORY: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdzD8KpTHz6_gsw9pPxRFlX
OTHER PLAYLISTS:
► Learning Math Series
https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw
►Cool Math Series:
https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho
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