25

Your question is a variant of the teabag problem.
I don't believe an exact answer is known, but
for the $1 \times 1$ square teabag, the maximum volume is about $0.2$:
(Image from Wikipedia article.)
The primary reference is Anthony Robin's 2004 article, "Paper Bag Problem".
...

answered Dec 31 '15 at 20:51

Joseph O'Rourke

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It is known (and follows from an easy calculation) that solutions of the isoperimetric problem on a surface have constant geodesic curvature. In his 1887 classic Leçons Sur La Théorie Générale Des Surfaces Et Les Applications Géométriques Du Calcul Infinitésimal, G. Darboux states, without indicating a proof, that a surface for which all of the curves of ...

answered Jun 26 '12 at 11:59

Robert Bryant

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19

There is probably no single proof that would provide a rigorous justification of the OP's principle in all cases. Moreover, without specifying more clearly what is meant by a 'natural map', the principle itself turns out not to hold in general.
For example, every smooth (complex-valued) function $f$ on the unit circle $S^1\subset\mathbb{C}$ can be ...

dg.differential-geometry fa.functional-analysis riemannian-geometry differential-equations calculus-of-variations

answered Apr 10 '15 at 1:54

Robert Bryant

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16

There are some differences. For example Bishop-Phelps theorem, which holds only in real Banach spaces. In my opinion, this qualifies as a "major theorem".
MR1749671
Lomonosov, Victor
A counterexample to the Bishop-Phelps theorem in complex spaces.
Israel J. Math. 115 (2000), 25–28.
Remark. Your statement "natural to assume that the field is real if the ...

answered Nov 26 '16 at 16:54

Alexandre Eremenko

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The answer seems to be $\frac{1}{2\pi}$, using a semi circle. See
Moran, P. A. P. "On a problem of S. Ulam." Journal of the London Mathematical Society 1.3 (1946): 175-179.

15

If you just calculate using the moving frame, you'll get the answer for the variation of the principal curvatures in a few lines:
$$
\delta\kappa_i = \mathrm{Hess}(u)(e_i,e_i) + \kappa_i^2\,u .
$$
Here, $\delta\kappa_i$ is the first $t$-derivative of $\kappa_i$ at $t=0$ (i.e., the 'first variation of $\kappa_i$'), $\mathrm{Hess}(u)$ is the quadratic form ...

answered Oct 7 '14 at 15:14

Robert Bryant

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Not exactly 'tweetable', but perhaps the identity (1) may help, if all you want to do is avoid the Euler-Lagrange equations. For simplicity, assume that $M^n$ is oriented. (One can write the identity (1) below as an identity on densities, so the orientability hypothesis is not essential, but I'll leave that detail for the interested.)
If $g$ is a metric ...

answered Feb 15 '17 at 15:16

Robert Bryant

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14

There is a coordinate-free description using only natural objects on $TM$. Here is one way to do it.
First, consider the basepoint submersion $\pi:TM\to M$. For each $v\in TM$, the linear map $\pi'(v):T_v(TM)\to T_{\pi(v)}M$ is surjective, and the $\pi$-fiber through $v$ is equal to $T_{\pi(v)}M$, a vector space. It follows that the kernel of $\pi'(v)$ ...

answered Apr 2 '13 at 17:19

Robert Bryant

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14

There is a book on the subject: "Information Theory and The Central Limit Theorem" by Oliver Johnson.
The article by Anshelevich mentioned by Yemon considers the operator $T$ acting on probability densities and corresponding to going from the law of a random variable $X$ to that of $(X+Y)/\sqrt{2}$ where $Y$ is an independent copy of $X$. The entropy is a ...

answered Oct 13 '14 at 14:13

Abdelmalek Abdesselam

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14

You are asking whether a least area surface depends continuously on its boundary curve, or in mathematical terms: whether for every curve $\gamma$ there exists a least area surface $S$ with $\partial S=\gamma$ such that for any neighborhood of $S$ in the space of surfaces there exists a neighborhood of $\gamma$ in the space of curves such that for any curve $...

dg.differential-geometry mg.metric-geometry calculus-of-variations geometric-measure-theory geometric-analysis

13

According to Giaquinta and Hildebrandt (Calculus of Variations I, p. 70): "Euler's differential equation was first stated by Euler in his Methodus inveniendi [2], Chapter 2, no. 21. Quite often, one speaks of Lagrange's differential equation, or the Euler-Lagrange equations. Yet Lagrange himself attributes this equation to Euler: 'Cette équation est quelle ...

answered Jul 31 '12 at 19:59

Robert Bryant

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13

Thanks for explaining your motivation, because I think that the general problem as you stated it is impossibly hard, but that, fortunately, for the problem that you are really trying to tackle (the inverse problem in the calculus of variations), there is no need to solve this problem in this generality. If you are willing to take advantage of the intrinsic ...

answered Jan 1 '13 at 0:41

Robert Bryant

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13

There's a 1985 article by Derriennic called "Entropie, theoremes limite et marches aleatoires" (entropy, limit theorems and random walks). In it there is a section where the connection between your observation that the Gaussian maximizes entropy (which is attributed to Shannon) and the central limit theorem is discussed.
He begins by discussing a proof (...

13

I do not know if it is good form for MO to cite one's own papers when answering a question, but I will take the chance. This matter is addressed in quite a bit of detail in my joint paper with Romeo Brunetti and Klaus Fredenhagen,
R. Brunetti, K. Fredenhagen, P. L. Ribeiro, Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics ...

answered Dec 27 '19 at 22:04

Pedro Lauridsen Ribeiro

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12

Now that your comment has clarified your question, we can answer it: The answer is 'no'. There is the following well-known example:
Consider the following family of circles: $C_\lambda$ is defined as $x^2+y^2 = 1$ and $z = \lambda$. Let $\lambda>0$ be fixed and orient $C_{-\lambda}$ counterclockwise and orient $C_\lambda$ clockwise. Then for $\...

answered Nov 21 '14 at 11:09

Robert Bryant

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12

I just bumped into your post after doing some google search to find a bibtex entry for Almgren's notes.
I have a copy of them: I could copy it and send it to you (or maybe scan it and share it via dropbox). However before getting to the combinatorial arguments you would have to go through quite a lot of material. I am writing a paper with a PhD student ...

12

There is no reason to believe that there is a supremum of this functional. For example, consider the $3$-torus $M = \mathbb{R}^3/\mathbb{Z}^3$
with the quotient metric and the unit $1$-forms
$$
\alpha_n = \cos(2\pi n z)\,\mathrm{d} x - \sin(2\pi n z)\,\mathrm{d} y,
$$
where $n$ is an integer, which are well-defined on $M$.
One finds by calculation that
$$...

answered Sep 11 '17 at 23:12

Robert Bryant

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11

I think if you take the metric on $\mathbb{R}^2$ obtained by rotating a curve which is $\sqrt{1-x^2}$ for $-1\leq x\leq 0$, and $x^2+1$ for $x\geq 0$ around the $x$-axis, then I think there will be a single closed contractible geodesic obtained by rotating the point $(0,1)$ around the $x$-axis.

11

Here's an example to show that the infimum is not always attained:
Consider the standard Hopf map $\pi:S^3\to S^2$, which is not null-homotopic, of course, so it follows that the area of the graph in $S^3\times S^2$ of any differentiable map $f:S^3\to S^2$ that is homotopic to $\pi$ is strictly greater than the area of the graph of a constant map, i.e., of $...

at.algebraic-topology dg.differential-geometry calculus-of-variations homotopy-groups-of-sphere global-optimization

answered May 10 '17 at 13:04

Robert Bryant

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11

There is a large literature on this, and the roots go back more than one hundred years. Some of the modern work along these lines can be found by looking for papers containing the term 'variational bicomplex'. For example, look at the papers and books by Ian Anderson and his group.
You can also look at papers and books by Phillip Griffiths and his ...

answered Jan 18 '13 at 21:35

Robert Bryant

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11

Actually this is one of the oldest problems in the calculus of variations. It's named "the Newton problem" after Sir Isaac Newton, who studied it in 1685. It arises from the determination of the optimal profile for the motion of bodies (projectiles, ships, etc), that is, the profile giving the minimal aerodynamic or hydrodynamic resistance. Here you are ...

10

This can be found in Besse "Einstein Manifolds", in chapter 4.
The idea is to use Koszul formula for the Levi-Civitta connection to compute the derivative of the curvature with respect to the metric. Bianchi identities also help.

10

The principle you mention is not always true ! V. Arnold proved that every continuous function in $N$ real variables is a composition of continuous functions of two variables only. More precisely, there exist $N(2N+1)$ universal functions $\phi_{ij}:[0,1]\rightarrow[0,1]$ such that the map
$$(g_1,\ldots,g_{2N+1})\mapsto\sum_jg_j\left(\sum_i\phi_{ij}(x_i)\...

dg.differential-geometry fa.functional-analysis riemannian-geometry differential-equations calculus-of-variations

9

A discussion of Tonelli's contributions and their relation to Hilbert's work can be found in this AMS bulletin. The original work was published in Italian, Fondamenti di Calcolo delle Variazioni (Bologna, 1921 & 1923) --- I have not found an English translation, but an English summary by Tonelli is also in an AMS bulletin. It is noteworthy that Tonelli ...

answered Mar 26 '14 at 9:57

Carlo Beenakker

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8

For any sensible definition of perimeter, an adaptation of following argument proves the claim. Consider the nearest-point projection map $p_A:\mathbb R^n\to A$ (that is, for every $x\in\mathbb R^n$, let $p_A(x)$ be the point of $A$ nearest to $x$). It is easy to see that $p_A$ is Lipschitz-1, i.e., $|p_A(x)-p_A(y)|\le |x-y|$ for all $x,y\in\mathbb R^n$. ...

8

The book by Gelfand and Fomin is quite good (and its Dover ...). Another one I like a great deal are those of Giaquinta and Hildebrandt (specially volume 1), but those are not Dover: check them out from the library!

8

Yes, this is called the nonlinear variation of constants formula due to Alekseev: “An estimate for the perturbations of the solutions of ordinary differential equations”, in: Vestnik Moskov. Univ. Ser. I Mat. Meh. 2 (1961), pp. 28–36. I don't think that that article is available in English.
It can also be found in the book by V. Lakshmikantham and S. Leela "...

ca.classical-analysis-and-odes mp.mathematical-physics sg.symplectic-geometry calculus-of-variations hamiltonian-mechanics

8

Your question actually is quite well answered around page 10 of Giaquinta and Hildebrandt's Calculus of Variations I: the Lagrangian formalism. The upshot is that the correct phrase you are looking for is the (possibly nonlinear) Gateaux differential, and not the Frechet derivative, and that is for good reason
(with my apologies to French people everywhere ...

8

In optimization, you typically have a function $f \colon X \to \mathbb{R}$ which you are going to minimize. In order to apply first-order optimality conditions or first-order methods, you would like to compute the derivative of $f$. However, if $X$ is a complex Banach space, the derivative will never be $\mathbb{C}$-linear, since the range of the derivative ...

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