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Mathematics Colloquium

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Date:  Tuesday, January 22, 2013

Title:  The Isoperimetric Problem Revisited: Extracting a Short Proof of Sufficiency from Euler's 1744 Proof of Necessity

Abstract:  Our primary objective in this talk is, with the student in mind, to present what we believe to be the shortest, most elementary, and most teachable solution of the isoperimetric problem in history. A secondary objective is to give a brief, but reasonably complete, overview of the remarkable life of the isoperimetric problem, and in the process demonstrate that it has been the most impactful mathematics problem of all time. In 1744 Euler constructed multiplier theory to solve the isoperimetric problem. However, contrary to Euler's belief, satisfaction of his multiplier rule is only a necessary condition and not a sufficient condition to demonstrate that the circle is the solution. Some 135 years later Weierstrass constructed his elegant sufficiency theory for problems in the calculus of variations and used it to provide what is accepted today as the first complete proof that the circle solves the isoperimetric problem. A multitude of sufficiency proofs ensued and in 1995 in a short paper aptly entitled A Short Path to the Shortest Path Peter Lax constructed what is considered to be the shortest and most elementary of all existing proofs. This background material is presented to set the stage for our demonstration that Euler's original necessity proof is but an observation away from establishing a sufficiency proof that we believe to be the shortest and most elementary in the history of the isoperimetric problem. We contemplate to what extend Euler or Lagrange could have, or should have, made our observation. Included is a contrast of our short proof with the Peter Lax short proof and an argument that historically the process of solving the isoperimetric problem was greatly compromised by the fact that the mathematicians of that golden era did not pursue functional convexity and the powerful optimization sufficiency theory that follows directly from this notion.


Speaker:  Richard Tapia
Institution:  Rice University

 

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