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I wouldn’t give a fig for the simplicity on this side of complexity. But I would give my right arm for the simplicity on the far side of complexity. -- Oliver Wendell Holmes

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The double bubble theorem in 3-space Every standard double bubble in R 3 has the least surface area required to separately enclose two volumes.

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Morgan Foisy Hass, Schlafly Hutchings Wichiramala Ritore, Ros Reichardt Morgan Foisy Hass, Schlafly Hutchings Wichiramala Ritore, Ros Reichardt

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A unified isoperimetric inequality Objects in R n having volume (or area) V and surface area (or perimeter) S satisfy the sharp inequality where r is the inradius of the minimizer in any of the following classes of objects:

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Classes for which this inequality holds include: All bodies (with minimizer the round ball) All rectangular boxes (with minimizer the cube) All triangles (minimizer is equilateral) Cylindrical cans (popular calculus problem)

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Classes for which this inequality holds include: All bodies (with minimizer the round ball) All rectangular boxes (with minimizer the cube) All triangles (minimizer is equilateral) Cylindrical cans (popular calculus problem)

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Double and triple bubbles in the plane Double bubbles in 3-space Conjecturally: All multiple bubbles in the plane Triple to quintuple bubbles in 3-space (n+1)-fold bubbles in R n

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Calibration is a great way to prove minimization. Find a progress monitor, in the form of a differential form or vector field Using a form of Stokes’ theorem to orchestrate the process, Make a (fully) local comparison between area and the integral of the monitor. The total monitor integral is the same for all competitors. Conclude the global comparison between competitors.

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Metacalibration can be described as calibration combined with slicing, and enhanced by emulation. Slicing makes possible new variable types, and can average out a pointwise inequality requirement over a curve or sub-surface. Emulation guides and simplifies the statement and calculations.

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Benefits of metacalibration are centered around the concepts of: Partial reduction Reallocation Emulation Differentiation of a measuring stick

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Emulation 1. Start with two objects to compare: An ideal object I A competing object C 2. Match some aspect of C and I 3. Measure some aspect of I, based on step 2 4. Use that quantity to help measure C

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To illustrate emulation, we offer the following isoperimetric proof, a-la-Schmidt: Theorem: For any body of volume V in R n, the surface area S satisfies where r is the radius of the round ball of volume V.

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Theorem: For any body C of volume V in R n, the surface area S satisfies Proof: The theorem is true if n=1, in which case V = L = 2r and S = 2. Now take any n>1 and assume the theorem true for n-1.

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Let C be a body of volume V in R n. Slice C with horizontal planes P t : {x n =t}. Let A max be the largest cross-sectional area. Let B be the round ball whose largest horizontal slice has area A max as well. Let V be the volume of B. C B

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C B Now as the slicing plane P t passes upward through C, for every t find a plane Q t slicing through B so as to match the cross-sectional area A(t). Let z(t) be the z-coordinate of the plane Qt, with z=0 at the center of B.

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C B Define G(t) = 2V(t) + V(t) - z(t) A(t) r Then G’ =2A + Az ’ - z ’A - z A’ r

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Define G(t) = (n-1)V(t) + V(t) - z(t) A(t) r Then G’ = (n-1)A + Az ’ - z ’A - z A’ r (n-1)A - z A’ r =

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Define G(t) = (n-1)V(t) + V(t) - z(t) A(t) r Then G’ = (n-1)A + Az ’ - z ’A - z A’ r (n-1)A - z A’ r = P - z A’ r by the induction hypothesis, where is the radius of the current slice of B.

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Define G(t) = (n-1)V(t) + V(t) - z(t) A(t) r Then G’ = (n-1)A + Az ’ - z ’A - z A’ r (n-1)A - z A’ r = G’ P - z A’ r by induction, where is the radius of the current slice of B. G’ ( - z) r 1 (P, A’)

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Define G(t) = (n-1)V(t) + V(t) - z(t) A(t) r Then G’ = (n-1)A + Az ’ - z ’A - z A’ r (n-1)A - z A’ r = G’ P - z A’ r by induction, where is the radius of the current slice of B. G’ ( - z) r 1 (P, A’) S’

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Define G(t) = (n-1)V(t) + V(t) - z(t) A(t) r G’ S’ G S In the end, G = [(n-1)V + V ] / r =V r V r V r V r + + … + + which, by the AM-GM inequality, is minimized when V = V and thus r = r. So

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G S In the end, G = [(n-1)V + V ] / r =V r V r V r V r + + … + + which, by the AM-GM inequality, is minimized when V = V and thus r = r. So completing the proof by induction.

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Comparison of methods for proving geometric minimization Deformation Variational methods Symmetrization Reduction Symmetrization Mod out by symmetry Directed slicing Calibration Equivalent problems Paired calibration

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Slicing

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Deformation Variational methods Symmetrization Reduction Symmetrization Mod out by symmetry Directed slicing Calibration Equivalent problems Paired calibration

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Paired vector fields

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Deformation Variational methods Symmetrization Reduction Symmetrization Mod out by symmetry Directed slicing Calibration Equivalent problems Paired calibration Metacalibration brings all these methods into one framework.

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Deformation Variational methods more localized Symmetrization more versatile Reduction Symmetrization Mod out by symmetry Directed slicing more flexible Calibration applicable to more types Equivalent problems a central feature Paired calibration less rigid Metacalibration brings all these methods into one framework.

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The double bubble theorem in 3-space 2 3

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What is the question to which this piece is the answer? Metacalibration

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What is the question to which this piece is the answer? Answer (i.e., question): Least “capillary surface area” for the given, fixed volumes

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Divide and conquer Partition into pieces… Partition into pieces… Solve planar problems via Hutchings Solve planar problems via Hutchings Coordinate these results overall slices Coordinate these results over all slices

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h1h1 h2h2 Slice competitor with horizontal planes Slice standard model with slanted planes, matching both volumes:

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h1h1 h2h2 Proof. Slice competitor with horizontal planes Slice standard model, matching both volumes: Prove that such slicing planes exist and are unique Prove that S’ ≥ G’, where G is the calibration

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h1h1 h2h2 Proof that S’ ≥ G’ uses variations equivalent problems calibration spherical inversion escorting Michael Hutchings’ planar method

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h1h1 h2h2

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Chapter 11-Functions of Several Variables

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