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**Dr. Richard Tapia Receives National Medal of Science**

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The Isoperimetric Problem Revisited: Extracting a Short Proof of Sufficiency from Euler’s 1744 Proof of Necessity Richard Tapia Rice University University Professor Maxfield-Oshman Professor in Engineering SIAM Annual Meeting Minneapolis, Minnesota July 11, 2012

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**Dedicated to Peter Lax in Recognition of his Numerous Mathematical Contributions**

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This talk and a paper with the same title, written in support of this talk, can be found on my website Just Google Richard Tapia

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**Optimization: The Cradle of Contemporary Mathematics**

Optimization problems are relatively easy to understand when compared with problems in many other branches of mathematics. Controversy invariably leads to interest. Hence, important optimization problems embedded in some controversy have played major roles in motivating and promoting mathematical activity. Mathematical, indeed scientific, activity can be motivated by many factors, and not all are removed from human emotion, as some might have us believe.

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Talk Objectives Promote the belief that the isoperimetric problem has been the most impactful mathematics problem in history Give a brief historical development of the isoperimetric problem and its solution. Demonstrate that Euler’s 1744 necessity proof is but an observation away from establishing a sufficiency proof that we believe to be the shortest, the most elementary, and the most teachable proof in history of the isoperimetric problem.

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Talk Objectives Contrast our proof with the currently accepted most elementary proof, that given by Peter Lax in 1995. Argue that the process of solving the isoperimetric problem was greatly compromised by the fact that mathematicians of the golden era 1630 – 1890 did not pursue functional convexity, it was a 20th century construct.

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**Outline The isoperimetric problem, its origins and promotion.**

Proofs: Zenodorus, Euler, Steiner, Weierstrass, Carathéodory, and Hurwitz. Lax’s short proof. Our short proofs. Mysteries from the Golden Era. Meet the players: Fermat, Euler, Lagrange, Steiner, and Weierstrass

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**The Isoperimetric Problem**

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**Queen Dido and the Isoperimetric Problem**

The Aeneid: written by Virgil in period BC Dido – Life in danger flees her homeland with wealth and entourage Finds new land and bargains with local king for a piece of land that she can mark out with the hide of a bull The Dido trick: cut hide into as many thin strips as possible to form a long cord, using the seashore as one edge, lay out the cord in the form of a semi-circle.

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**Dido Purchases Land for the Foundation of Carthage**

Dido Purchases Land for the Foundation of Carthage. Engraving by Matth¨aus Merian the Elder, in Historiche Chronica, Frankfurt a.M., Dido’s people cut the hide of an ox into thin strips and try to enclose a maximal domain.

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**Application of the Dido Maximum Priciple**

Medieval map of Cologne

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**Yet Another Application of the Dido Maximum Principle**

Medieval map of Paris

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The Early Greeks II Zenodorus (Greek mathematician who lived BC) Wrote work entitled On Isometric Figures Work lost, but referenced by Pappus and Theon 300 years later Studied figures with equal perimeters and different shapes Proved The circle has a greater area than the regular polygon on the same perimeter Of two regular polygons of the same perimeter, the one with the greater number of angles has the greatest area Stated as scolia In 2-D the circle solves the Isoperimetric Problem In 3- D the sphere solves the Isoperimetric Problem Can not demonstrate that he gave an incorrect or incomplete proof as many historians believe

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Euler (1744) II

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Euler (1744) II Goldstine states: It is interesting that Euler did not completely understand the fact that his condition[satisfaction of the Euler-Lagrange equation] is a necessary but not a sufficient one. In his discussion it is clear that he felt his condition was sufficient to ensure an extremum, and that by evaluating the integral along an extremal [and one other curve] he could decide whether it was s a maximum or a minimum.

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**Euler General Isoperimetric Problem**

II

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**Queen Dido form of the Isoperimetric Problem**

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**Euler’s 1744 Proof of Necessity as Sufficiency**

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Steiner (1838) II Jakob Steiner ( AD) was one of the most brilliant and creative geometers in history. His mathematical inquiries were confined to geometry to the total exclusion of analysis. In fact he hated analysis and doubted if anything that could not be proved with geometry could be proved with analysis. In Steiner gave the first of his five equivalent proofs that the circle solved the isoperimetric problem. His proofs used synthetic geometry and were mathematically quite elegant. Mathematical historians embrace and promote his proofs and call them models of mathematical ingenuity. They became very visible in the mathematical community. Steiner boasted that he had done with geometry what had not been done,

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Steiner (1838) II and could not be done, with analysis, i.e., solve the isoperimetric problem. What Steiner proved was that any curve which was not the circle could be modified using a geometric procedure now called Steiner Symmetrization to obtain a curve with the same perimeter but a larger area. He then concluded that as a consequence the circle must be the solution to the isoperimetric problem. As such he fell into the use of necessity as sufficiency trap and made the trap rather infamous. The analysts of the time, led by Peter Dirichlet, pointed out to Steiner that his proof is not valid unless he assumes that the isoperimetric problem has a solution, i.e. existence. Steiner did not accept the criticism well, and rebutted with a very

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Steiner (1838) II superficial argument that he claimed demonstrated that the isoperimetric problem must have a solution. Actually, at best he demonstrated that existence of an upper bond for the area functional. Now analysts observed that if the Steiner symmetrization process could be applied repeatedly creating a sequence of curves with the same perimeter but increasing area that converges to the area of the circle, the so-called process of completing Steiner’s proof, then the flaw in Steiner’s proof would be removed.

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**The Steiner 3 Step Proof Step1 The curve must be convex:**

Step2 Perimeter bisector divides the curve into equal areas and we can therefore symmetrize across the bisector

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**Preliminary Results from Geometry needed to complete Steiner’s Proof**

Lemma 1 (Thales Theorem 600 BC) If AC is a diameter, than the angle at B is a right angle. Lemma 2 (Converse to Thales Theorem) A right triangle’s hypotenuse is a diameter of its circumcircle. Lemma 3 Of all possible triangles with two sides of specified lengths the triangle of maximum area is the right triangle.

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**The Steiner 3 Step Proof continued “STEINER SYMMETRIZATION”**

Step 3 All inscribed angles determined by the perimeter bisector must be right angles.

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Weierstrass (1879) II Concerning the solution of the isoperimetric problem, Weierstrass was quite aware of the shortcomings of Steiner’s proof and somewhat bothered by Steiner’s arrogantly promoted negative view of analysis and analysts. Hence he boldly and proudly placed himself in the noble role of defender of analysis and vowed to solve the isoperimetric problem using analysis. He introduces his work with the following statements concerning the solution of the isoperimetric problem:

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Weierstrass (1879) II “A detailed discussion of this problem is desirable, since Steiner was of the opinion that the methods of the calculus of variations were not sufficient to give a complete proof, but the calculus of variations is in a position to prove all this, as we will show later; furthermore it can show what Steiner could not – that such a maximum really exists.”

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Weierstrass (1879) II Weierstrass first builds an elegant and sophisticated sufficiency theory for the simplest problem from the calculus of variations employing such subtle notions as Jacobi’s notion of conjugate points and his own notion of fields of extremals. He then extends this sufficiency theory to the isoperimetric problem by turning to Euler’s multiplier rule and applying his sufficiency theory to the Euler’s auxiliary problem. Using this theory he demonstrates that Euler’s auxiliary problem for the isoperimetric problem has the circle as solution; hence the isoperimetric problem has the circle as solution. So, according to the literature, some 135 years after

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Weierstrass (1879) II Euler’s proof of necessity we have the first sufficiency proof. While this notable work gave the world its first sufficiency proof for the isoperimetric problem, we expect to convince the audience that Weierstrass really used a sludge hammer to pound a nail. His sophisticated sufficiency theory is not needed to merely demonstrate that the circle solves Euler’s auxiliary problem.

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**II Interesting Post – Weierstrass Proof**

Carathéodory’s (1910) completion of Steiner’s Proof.

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**II Interesting Post – Weierstrass Proof**

Hurwitz (student of Weierstrass) 1902 proof using Fourier series.

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**“A Competition for the World’s Most Elementary Solution of the Isoperimetric Problem”**

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**Pedro Lax y Ricardo Tapia “Mano a Mano”**

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The Winner!

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**Now on to more serious things like Mathematical Proofs**

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**A Short Proof of Sufficiency Given by**

Peter Lax (1995) III

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**IV Our Short Proof of Sufficiency Motivated by Euler**

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Question: Could Euler have made our observation at the time of his 1744 writing? The foundation of our observation is Taylor’s theorem with remainder. The literature tells us that Taylor published his theorem in So Euler most likely was aware of Taylor’s theorem in However, the rub is that Taylor’s theorem with remainder was not known at that time. It is somewhat ironic that the form of the remainder that we used in our proof is credited to Lagrange in 1797, and is actually referred to today as the Lagrange form of the remainder. So Euler would not have been in good position to make our observation.

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**IV Our Short Proof of Sufficiency Motivated by Lagrange**

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Remark Lagrange could have made this proof because he was familiar with the form of Taylor’s theorem that we used, indeed it is due to him. While this hypothesized proof would have been made 50 years after Euler, it would still have been some 80 years before Weierstrass.

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**Mysteries from the Golden Era (1630-1890)**

Euler thinking that all extremals were either minimizers or maximizers. The use of necessity as sufficiency Lagrange not attempting to solve the isoperimetric problem

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**The Greatest Mystery from the Golden Era**

The failure to develop functional convexity and consequently the powerful optimization sufficiency theory that follows Historical Development of Convexity Archimedes (287 BC – 212 BC) On the Sphere and the Cylinder A convex arc is a plane curve which lies on side of the line joining its end points and all cords of which lie on the same side. Fenchel’s Explanation for the failure to develop functional convexity When in the seventeenth century Archimedes methods were taken up again, convexity played still a role, for instance in the work Fermat. But the development of the calculus let them recede to the background and even be forgotten.

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**Golden Era Uses of Convex Arc (Not Convex Function)**

Fermat 1630 Steiner 1838 Cauchy 1850 Minkowski (1910) in his book initiates the field of convex geometry and promotes the role of convexity in mathematics.

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**Introduction of Convexity**

First Definition of A convex function (on interval of reals) – Jensen (1905) A general convex set – Steinitz (1913) Use of convexity in optimization followed linear programing contributions from 1940’s Why so late?

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**The Use of Convexity in Optimization Theory Today**

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**The Use of Convexity in Optimization Theory Today**

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**Convexity Observations**

The isoperimetric problem is not a convex program But its Euler Multiplier Rule formulation is a convex program. The iso-area program is a convex program. For purposes of illustration lets follow this path.

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Conclusions Isoperimetric problem is history’s most impactful mathematical problem Our short proof is most elementary and most teachable. Effective solution of the isoperimetric problem was seriously delayed because the great minds of the Golden Era ( ) did not develop functional convexity.

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**(in order of performance)**

Lets meet the players (in order of performance)

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**A Snapshot of Fermat “The Prince of Amatuers”**

Lived 1601–1665 Born and educated and functioned as a lawyer and legal oﬃcial in Toulouse, France Did mathematics for recreation and considered it a hobby. He dabbled and rarely produced proofs. As such he was sloppy and chose not to include detail or polish his work. Much of his work required “a ﬁll in the blanks” activity as is characterized by his last theorem. He did not publish and communicated his work in the form of letters to important people. Directly and naively engaged in controversy with prominent mathematicians of the times and in particular with the powerful Descartes.

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Conceived and applied the diﬀerential calculus in a 1628 unpublished work entitled Minima and Maxima and the tangent to a curve. Note that this was 15 years before Newton was born and 18 years before Leibniz was born. Ten years later in 1638 he made his work semi-public in a letter to Descartes. The work stepped strongly on the toes of work that Descartes was doing on tangents to curves. When asked for an oﬃcial evaluation of Descartes work he wrote “he is groping around, in the shadows.” Descartes responded with the public statement “Fermat is inadequate as a mathematician and as a thinker.” This damaged Fermat’s reputation. In prominent competition between Descartes and Fermat on the notion of a tangent to a curve, Fermat won and Descartes lost. After the dust settles Descartes writes to Fermat “your work on tangents is very good, if you had explained it well from the onset, I would not have had to criticize it or you.”

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Fermat had little to no interest in physical applications of mathematics, he just loved the math. This is in strong contrast to Newton who did the mathematics for his love of physical applications. Even his notation and terminology showed this.

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A snapshot of Euler Born in Switzerland in Died in Russia in 1783 at the age of 76 Clearly the most proliﬁc mathematician the world has ever known. Laplace -“Read Euler, read Euler, he is the master of us all.” Very fond of children. He had 13, but 8 died at early ages. He could work anywhere, under any conditions. It is said that he once wrote a math paper in the 30 minutes before dinner with one child on his lap and others running around his chair. Phenomenal memory and mental calculation power. He memorized the entire Aeneid.

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**Had sight problems his entire life**

Had sight problems his entire life. Was blind the last 17 years of his life. These years were very productive. During this time period he developed the lunar theory, something that escaped Newton. All the necessary analysis and calculation was done in his head. Very supportive of other mathematicians, especially the young Lagrange. He dropped his multiplier theory in favor of that of Lagrange and he called it Lagrange Multiplier Theory He was a devout Christian and argued with the prominent atheists of the time. Since Euler knew no philosophy, Voltaire tied him into metaphysical knots and made great fun of him in Fredericks Court. Euler himself laughed when others laughed at him

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A snapshot of Lagrange Born in Italy to wealthy parents in Died in France in 1813 at the age of 77. Quite precocious and at the age of 23 he was considered one of the greatest mathematicians of the time including Euler and the Bernoullis. During his early years, his parents lost their wealth, In later years Lagrange said that this was good for him, “If I had inherited a fortune, I would not have cast my lot with mathematics.” Generously appreciative of the work of others, and dissatisﬁed with his own. Considered his early work on the calculus of variations performed at the age of 19 his best.

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Befriended Napolean Married for social convenience, but grew very fond of his wife. Around the age of 50 he had become quite disillusioned with mathematics. At 51 his wife died. He became depressed and did nothing for 5 years. At age 56 he married the 17 year old daughter of a colleague. His excitement for math returned and very productive years followed. She rescued him from his twilight between life and death. Honors were showered on him by the French. Concerning these awards he said my greatest trophy is having found such a tender and devoted companion as my young wife. Here he probably coined the phrase “trophy wife.”

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A snapshot of Steiner (1796 – 1863) Born and raised on a farm in Bern Switzerland. He was extremely poor, did not go to school and did not learn to read or write until the age of 14. At the age of 22 Steiner took classes in mathematics at the University of Heidelberg. He supported himself by tutoring in mathematics. At the age of 26 he decided to move to the more prestigious University, The University of Berlin. There he befriended Abel, Jacob, and Crelle of Crelle’s Journal. Leaving the University two years later, Steiner found a post at a Berlin technical school and spent the next 10 years teaching mathematics there. He taught basic concepts to the young students and the working atmosphere was not good he wrote and published many significant paper, mostly in Crelle’s Journal. He became the world leader in projective geometry.

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At the age of 37 he was awarded an honorary doctorate by the University of Königsberg on the recommendation of Jacobi At age of 38 he was elected to the Prussian Academy of Sciences, again on the recommendation of Jacobi and other leading mathematicians. At the the age of 38 he was appointed to a new extraordinary professorship of geometry at the university of Berlin, again on the recommendation of Jacobi and other. Steiner remained an extraordinary professor at the university until his death 29 years later. He never married, but dedicated much of his adult life to his students. A blunt and somewhat coarse manner combined with surprisingly liberal social attitudes made Steiner a unique individual, particularly to students. He influenced many of his students, including the noted mathematician Georg Friedrich Bernhard Riemann.

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**A Snapshot of Weierstrass**

Born in Germany in 1815 Died in Germany in 1897 at the age of 82 Excelled in High school despite having to work part time as a bookkeeper At the age of 19 his father sent him to university of Bonn to study commerce and law. He did not attempt either, he devoted his time to fencing, drinking beer, and studying math on his own. After 4 years he returned home without a degree At age 26 started teaching high school

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At the age of 38, as a high school teacher, he wrote a paper on Abelian functions that created a sensation. At the age of 41 he was awarded an assistant professorship at the University of Berlin where he became an outstanding lecturer, teacher, and mentor

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Weierstrass Weierstrass, in addition to his numerous mathematical contributions, is known for introducing rigor of proof and cleanliness of definition into the calculus of variations at a time that it was sorely needed. He did not publish his work in this area but developed a complete and polished set of lecture notes that he used in his university courses at the University of Berlin. Today we know about his many contributions in the calculus of variations from his collected works which was constructed primarily from the lecture notes of his numerous students during the time period 1865 – It is alleged that Weierstrass had 40 or so students during this time period and may of these students became quite distinguished in their own right; for example Cantor, Frobenius, (Sofia) Kowalewski ( as a women she was not officially accepted as a student at the University of Berlin and was given an honorary degree), Mittag-Leffler, Runge, Schur, and Schwarz.

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Weierstrass Weierstrass’ critical sense and need to base his analysis on such a firm foundation led him to continually revise and perfect his writings to the point that publication was precluded. In spite of this his work he became so well known that today he is called the father of analysis.

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**A Story of Mystery and Intrigue**

“Karl Weierstrass and Sofia Kowalewski” In 1870, at the age of 20, Sofia Kowalewski came to Berlin to study with Weierstrass who was 55 years old at the time. He taught her privately since she was not allowed admission to the University. She was an unusually gifted young woman in so many ways, Weierstrass: Sofia is my favorite pupil and my “weakness”. She died at the age of 41, he burnt all of her correspondence.

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**A Story of Mystery and Intrigue**

I would love to see a book written focusing exclusively on the Weierstrass-Kowalewski Relationship In fact I have a title for this book.

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**The Integral and the Integrand: A Story of Love and Respect**

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**Thank you for your attention.**

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