# Fun with simple analysis problems I: the rest of the story

In an earlier post with the same title (and without the subtitle) I introduced some thoughts that were triggered by this simple problem:

Suppose that

$|\frac{df}{dx}(x)| \leq \lambda |f(x)|$                    (1)

for all $x$, that $f$ is continuous and differentiable, and that $f(0) = 0$.

Prove that $f(x) = 0$ everywhere.

In that post (which you can find here  Fun with simple analysis problems I ),  I started by presenting three solutions and then generalized and explored further.

What I did not reveal in that post, was that writing it, gave me an idea for a more advanced problem. Not too long afterwards, Laramie Paxton joined my group and I gave him this problem to work on for his dissertation. We collaborated in solving the problem, since that is how I mentor all my students — their dissertations are collaborations with me. This resulted in a paper we wrote together: A Singular Integral Measure for $C^{1,1}$ and $C^1$ Boundaries that can be found here.

Laramie Paxton arrived at WSU quite naive with respect to analysis, having completed an online masters in mathematics that did not give him a good foundation in analysis. But he very quickly he adopted habits that led to rapid progress. He started by studying intensely the summer before arriving and passing the qualifying exam on his first try.  Then he took my challenging undergraduate analysis course (I used Fleming’s Functions of Several Variables), pushed through courses in advanced analysis, and geometric measure theory, and worked on applications in image analysis (generating papers he actually led) and finished his dissertation, all in the space of two years. After a year of postdoc, he landed the job he is about to start, at Marian University in Wisconsin. I believe that both the University and Laramie are lucky to have each other.

In general, I believe that small universities are good places to be nowadays, but from everything I hear, this place is better than good — it is perfect for Laramie’s talents and skills. (In addition to his impressively growing mathematical skills, he was already phenomenally skilled in logistics and organization which can be seen in his highly effective help in making the events listed here, from April 2017 to July 2018, a reality.)

A major point of both the original post on the problem and this present post, is that the paper with Laramie, as well as the results in the first post, flowed from taking time to think about a simple analysis problem that would usually be viewed as a not-too-hard exercise, not worthy of more thought than it takes to find one solution.

While I am sure that there are other undiscovered aspects of the problem that launched these two posts and Laramie’s dissertation problem, I believe that what has been explored illustrates why it makes sense to treat simple problems as invitations to playful exploration and creativity.

# Median Shapes

When I wrote the paper with Simon Morgan pointing out the $L^1\text{TV}$ functional was actually computing the flat norm for boundaries, we suggested this gave us a computational route to statistics in spaces of shapes. While earlier work certainly touched on this idea of using the flat norm for inference in shape spaces — see this paper on shape recognition, it was not until my student Yunfeng Hu collaborated with myself and Bala Krishnamoorthy (my collaborator, also a co-mentor of Yunfeng’s), that we started addressing the idea of statistics in shape spaces in the original paper with Simon Morgan.

The results can be found here: https://arxiv.org/abs/1802.04968 , in a paper with the title Median Shapes, with authors Yunfeng Hu, Matthew Hudelson, Bala Krishnamoorthy, Altansuren Tumurbaatar, Kevin R. Vixie. Tumurbaatar wrote the first complete version of the code used, and Matthew Hudelson contributed a pivotal new result on graphs inspired by a problem in the paper, while Bala led the computational end of things and I led, in collaboration with Yunfeng Hu (and Bala keeping us honest!), the theoretical parts of the paper. It was a fair bit of work.

We went over the more difficult results a few times, finding improvements and corrections. Of course, there may be a few things here and there to improve, but for now, it is done.

Yunfeng probably spent the most time writing up the piece proving that near regular points on the median, the collection of minimal surfaces meeting the median have a tangent structure we describe as a book.  While this is clear to experienced geometric analysts,  there are lots of little details and we wanted most of the paper to be more accessible to a wider audience. There are lots of other pieces here and there that took time to think about and write up (and rewrite). For example, when showing the set of medians need not contain any regular members,  the part where we show that we need only consider graphs when searching for a minimizer was not easy. And of course, as in most all of geometric analysis, there are problems you solve without too much effort at a high level, but find that writing down is tedious, though at times enlightening due to the fact that those little details turn out to be hard and illuminating.

Because the problem of computing the median reduces to a linear program, while the mean reduces to a quadratic program, we focused on the median problem. Some parts of the paper are a bit long winded, for the reason that we wanted it to have more details that would usually be in a paper communicating to others that understand geometric analysis.

Anyway, have a look. If you find yourself interested, there is already code you can use to compute medians, though we hope eventually to have faster code.

# An Invitation to Geometric Measure Theory: Part 1

While there are a variety of article-length introductions to geometric measure theory, ranging from Federer’s rather dry AMS Colloquium Talks to Fred Almgren’s engaging Questions and Answers to Alberti’s Article for the Encyclopedia of Mathematical Physics, I will take a different approach than has been taken in any of these and introduce geometric measure theory through the vehicle of the derivative.

### The Derivative, Geometrically

The derivative that is encountered for the first time in calculus is defined as the limit of a ratio of the “rise” over “run” of the graph of a function. For $y = f(x)$, this becomes

$\frac{df}{dx}$$(a)=\lim_{x\rightarrow a}$$\frac{f(x) - f(a)}{x-a}$.

This is visualized as the slope of the secant lines approaching a limit – the slope of the tangent line – as the free ends of those lines approach $(a,f(a))$. This is illustrated in the first figure.

The derivative as $\hat{L}_a$, the optimal linear approximation to f at a, is another, very useful way to think about the derivative. Here, we focus on the fact that the tangent line at $(a,f(a))$ approximates the graph of $f(x)$ at $(a,f(a))$ as we zoom in on the graph. More precisely, writing $x = h+a$,

$f(x) = f(h+a) = f(a) + \hat{L}_a(h) + g(h)h$,

where $\hat{L}_a$ is linear in $h$, $g(h)\rightarrow 0$ as $h \rightarrow 0$, and the tangent line L is the graph of the function $y = f(a) + \hat{L}_a(x-a)$ .

Exercise: use the facts that (1) linear $\hat{L}_a:\Bbb{R} \rightarrow \Bbb{R}$ have the form $h\rightarrow sh$, $s$ a scalar, and (2) $g(h) \rightarrow 0$ as $h \rightarrow 0$, to rearrange this last equation for $f(x)$ into the original definition of a derivative.

Using the equation above to get

$\left|f(x) - (f(a) + \hat{L}_a(x-a))\right| \leq (\sup_{|s|\in[0,\epsilon]} |g(s)|)|h|$ for  $h\in[-\epsilon,\epsilon]$,

we are able — after some work (see the exercise below) — to get this nice geometric interpretation:

The figure illustrates the fact that the graph of $f(x)$ lies in cones centered on $L$, whose angular widths go to zero as we restrict ourselves to smaller and smaller $\epsilon$-balls centered on $(a,f(a))$. Inside the $\epsilon_1$-ball, the graph stays in the wider cone, while in the smaller, $\epsilon_2$-ball the graph stays in the narrower cone.

Let’s restate this. Defining

1. $p \equiv (a,f(a))$,
2. $B(\epsilon)$ to be the ball of radius $\epsilon$ centered on $p$,
3. $F\equiv\{ (x,y) | y= f(x) \}$,
4. $C_L(p,\epsilon)$ to be the smallest closed cone, symmetrically centered on $L$, with vertex at $p$ such that $F\cap B(\epsilon) \subset C_L(p,\epsilon)$, and
5. $\theta (\epsilon)$ to be the angular width of $C_L(p,\epsilon)$,

we have that

f is differentiable at $a \Leftrightarrow \theta(\epsilon) \rightarrow 0\text{ as }\epsilon \rightarrow 0$

Here is a figure illustrating this:

Exercise: provide the missing details taking us from the above inequality bounding the deviation from linearity to the above statement that {f is differentiable at $a \Leftrightarrow \theta(\epsilon) \rightarrow 0\text{ as }\epsilon \rightarrow 0$} using the facts that (1) the above inequality defines cones that are almost symmetric about $L$ and (2) the $\epsilon$-ball centered at p is contained in the vertical strip $(x-a,y-f(a)) \in [-\epsilon,\epsilon] \times(-\infty,\infty)$

With this shift to a geometric perspective, we are now in a position to take a step in the direction of geometric measure theory.

Note that in our definition the cones contain all of the graph as they narrow down and we zoom in. But what if all we know is that a larger and larger fraction of the graph is in a narrower and narrower cone as we zoom into p? That is precisely the idea that approximate tangent lines capture. We will introduce two different versions of the concept.

### Densities as a path to an approximate tangent line

#### Tangent Cones

The tangent line discussed above is also the tangent cone. The tangent cone of a set in $\Bbb{R}^n$ can have any dimension from 1 to n. For nicely behaved k-dimensional sets, the tangent cone will also be k-dimensional. In the case of the usual derivative of functions from $\Bbb{R}$ to $\Bbb{R}$, we are working in the graph space $\Bbb{R}^2$ with 1-dimensional sets. Moving to tangent cones, we can approximate one dimensional sets which are not graphs or, more generally, arbitrary subsets of $\Bbb{R}^n$.

We now define the tangent cone of $F\subset\Bbb{R}^n$ at $p$.

To obtain the tangent cone, begin by translating $F$ by $-p$. (This moves $p$ to 0.) Define $F(\epsilon) \equiv (F\cap B(\epsilon))\setminus p$. Use a projection center at 0 to project the translated $F(\epsilon)$ onto the sphere of radius $\epsilon$ centered on 0. Take the closure of the resulting subset of the $\epsilon$-sphere. Finally take the cone over this set. Call this set $T_p^\epsilon(F)$. That is,

$T_p^\epsilon(F)=\{\Bbb{R}\geq 0\}(\text{Closure}($$\cup_{x\in F(\epsilon)}\frac{x-p}{|x-p|}$$)).$

Now define the tangent cone of F at p to be the intersection of $T_p^\epsilon(F)$ at any sequence of $\epsilon_i$‘s going to zero; $\epsilon_i = \frac{1}{i}$ will do. Thus the tangent cone of $F$ at p, $T_p(F)$ is given by:

$T_p(F)=\bigcap_i T_p^\frac{1}{i}(F).$

Here is a figure illustrating the key idea:

Note: the tangent cone is centered on the origin, 0, but I will be plotting it as though it were centered on p. Similarly, the tangent lines will sometimes be thought of as linear subspaces (i.e. centered on the origin 0, and other times as the shift of that linear subspace to p.

In the case of a differentiable function $f:\Bbb{R}\rightarrow\Bbb{R}$, this tangent cone is the usual 1-dimensional tangent line.

#### Densities

Now we need $\theta^k(\mu,F)$, the k-dimensional density of F at p.

Define $\omega(k)$ such that it agrees with the volume of the unit ball in $\Bbb{R}^k$ when k is an integer (there is a standard way to do this using $\Gamma$ functions). Let $\mu$ measure k-dimensional volume. Typically this will be k-dimensional  Hausdorff measure, $\mathcal{H}^k$. Whatever intuitive idea you have of k-dimensional measure is good enough for our purposes. (At the end of this post I also define Hausdorff measures more carefully.)

Now, $\theta^k(\mu,F)$ is given by

$\theta^k(\mu,F)=\lim_{\epsilon\rightarrow 0}$$\frac{\mu(F\cap B(\epsilon))}{\omega(k)\epsilon^k}$

when this limit exists. When the limit does not exist, we work with the limsup and liminf of the right hand side which are called upper and lower densities of F at p and are denoted by $\theta^{*k}(\mu,F)$ and $\theta^k_*(\mu,F)$ respectively.

#### Approximate Tangent Cones

We now define the approximate tangent cone at p to be the intersection of closed cones whose complements intersected with F have density zero at p:

$\tilde{T}_p(F)=\bigcap\{\text{closed cones }C\text{ with vertex }p|\theta^k(\mu,(\Bbb{R}^n\setminus C)\cap F)=0\}$

Originally (in this section), we were aiming at having a definition of approximate tangent line that was invariant to (small) pieces of the set F outside the sequence of cones, provided those pieces got small enough, quick enough. Now we can make that more precise. We want a definition of approximate tangent line that ignores such excursions of F provided these excursions have density zero at p. Rather anti-climatically then, here is the definition we have been waiting for (though you might have already guessed it!)

A 1-dimensional set has an approximate tangent line at $p$ when the approximate tangent cone is equal to a line through p.

When the curve is an embedded differentiable curve, the tangent line and the approximate tangent line are the same.

Remark: in general, when we are dealing with k-dimensional sets in $\Bbb{R}^n$, we will get approximate tangent k-planes.

Exercise: can you create examples of one dimensional sets which have a (density based) approximate tangent line at p but not the usual tangent line at p?

Exercise: prove that a tangent line to a continuous curve is also the (density based) approximate tangent line at p.

### Integration as a path to an approximate tangent line

There is different version of approximate tangent k-plane based on integration. (The one dimensional version is of course an approximate tangent line.)

We start with the fact that we can integrate functions defined on $\Bbb{R}^n$ over k-dimensional sets using k-dimensional measures $\mu$ (typically $\mathcal{H}^k$). We zoom in on the point p, through dilation of the set F:

$F_\rho(p) = \{x\in\Bbb{R}^n | \;\;x=\frac{y-p}{\rho}+\text{ p for some }y\in F\}.$

We will say that the set $F$ has an approximate tangent k-plane $L$ at p if the dilation of $F_\rho(p)$, converges weakly to $L$: i.e. if

$\int_{F_\rho} \phi d\mu\rightarrow_{\rho \rightarrow 0} \;\; \int_L \phi d\mu$

for all continuously differentiable, compactly supported $\phi:\Bbb{R}^n \rightarrow \Bbb{R}$.

In the next two figures, we illustrate this for the case of 1-planes – i.e.lines: in the first figure, $L$ is the weak limit of the dilations of F, while in the second it is not.

Note: solid green lines are the level sets of $\phi$ while the dashed green line indicates the boundary of the support of $\phi$. Note also that the $\rho$‘s of 0.4, 0.1, and 0.02 are approximate.

Exercise: can you create an example of a one dimensional curve which has the usual tangent line at p but not an (integration based) approximate tangent line at p?

### Closing Note On Hausdorff Measure

We would like a notion of k-dimensional volume or k-dimensional measure. In many cases, the right notion turns out to be k-dimensional Hausdorff measure. We already know what 1,2, and 3-dimensional measure is as long as the objects we are measuring are regular enough, like subsets of lines, rectangles, and cubes. It does not seem too much of a stretch to think that we can extend these measures to things that are somewhat wiggly. That is, we can still easily imagine measuring the length of a subset of a smoothly turning curve, or the area of a piece of a surface that undulates slowly. Hausdorff measure permits us to measure not only such smooth sets (giving the same result as any reasonable extension of the usual Lebesgue measures to the nice cases), but also to measure very wild sets (like fractals).

How to compute the k-dimensional Hausdorff measure of $A\subset \Bbb{R}^n$:

1. Cover A with a collection of sets  $\mathcal{E}= \{E_i\}_{i=1}^\infty$, where $diam(E_i) \leq d \;\; \forall i$. Here, $diam(E_i)$ is the diameter of $E_i$.
2. Compute the k-dimensional measure of that cover: $\mathcal{V}_\mathcal{E}^k(A) = \sum_i\omega(k) (\frac{diam(E_i)}{2})^k$
3. Define $\mathcal{H}_d^k(A)=\inf_{\mathcal{E}} \mathcal{V}_\mathcal{E}^k(A)$ where the infimum is taken over all covers whose elements with maximal diameter d.
4. Finally, we define: $\mathcal{H}^k(A)=\lim_{d\downarrow 0}\mathcal{H}_d^k(A).$

Remark: Suppose that for any  $\epsilon > 0$, there is a cover $\{E_i\}_1^\infty$ of $A$, such that $\sum_i diam(E_i) < \epsilon$. Then for $k \geq 1$, $\mathcal{H}^k(A) = 0$.

Here is a figure illustrating Hausdorff measures:

Clearly, this can be difficult to compute. It turns out though that in $\Bbb{R}^k$, $\mathcal{H}^k = \Bbb{R}^k$. And by use of mappings, this can take us quite a ways in computing $\mathcal{H}^k(A)$ for integral k and rather general $A$.

Exercise: Show that if $0 < \mathcal{H}^\gamma(A) < \infty$ then $\mathcal{H}^\alpha(A) = \infty$ and $\mathcal{H}^\beta(A) = 0$ for $\alpha < \gamma < \beta$.

# Geometric Measure Theory by the Book

There are an armful of texts that I have used to learn and teach geometric measure theory. In this note, I will give a review of these texts, which are:

1. Herbert Federer’s Geometric Measure Theory
2. Frank Morgan’s Geometric Measure Theory: A Beginner’s Guide
3. Krantz and Parks Geometric Integration Theory
4. Lin and Yang Geometric Measure Theory – an Introduction
5. Leon Simon’s Lectures on Geometric Measure Theory
6. Pertti Mattila’s The Geometry of Sets and Measures in Euclidean Spaces
7. Evans and Gariepy’s Measure Theory and Fine Properties of Functions
8. Ambrosio, Fusco, and Pallara’s Functions of Bounded Variation and Free Discontinuity Problems
9. Enrico Giusti’s Minimal Surfaces and Functions of Bounded Variation

My two favorites are Leon Simon’s Lectures on Geometric Measure Theory and Evans and Gariepy’s Measure Theory and Fine Properties of Functions. Before I dive into comments on each of the books, here is a bit of history concerning my path into the subject.

### The Backstory

I was first turned on to geometric measure theory by David Caraballo, the last student to finish with Fred Almgren before Fred died. (Fred who was famous for his deep results in geometric measure theory, was a student of Federer and a professor at Princeton.) I met David at the Nonlinear Control Theory Short Course, organized by Hector Sussmann and Kevin Grasse, at the 1999 Joint AMS-MAA meetings in San Antonio. David and I became instant friends and I was soon swept away by David’s passion for geometric measure theory, realizing that this field was also particularly well matched with my mathematical muses.

In particular, David talked up Evans and Gariepy’s text, so the first thing I did when I got back to Los Alamos was hibernate in my office and immerse myself in that text. It was beautiful and exhilarating — and I was hooked.

At the time I was working on inverse problems and dynamical systems. The inverse problems involved images and it was the image analysis that drew me further into geometric measure theory. The biggest influence in this migration was of course the rising prevalence of total variation regularization in image analysis methods. The first papers I read were David Strong’s. These inspired me to introduce total variation regularization into the sparse tomographic reconstructions methods we were working on. (There were other people dabbling with these methods at the same time at Los Alamos, but Tom Asaki and I took these methods and ran with them. Abel inversions were the tomographic workhorse at Los Alamos, so this was one of the first targets of our work. The first papers can be found here: Abel inversion using total-variation regularization and Abel inversion using total variation regularization: applications)

Through my work in image analysis, I met Andrea Bertozzi, and while visiting Duke at her invitation early in 2003, I met Bill Allard. Bill and I started a close collaboration on data analysis work at Los Alamos, and I began picking up pieces of geometric measure theory from him. Bill was a student of Fleming’s at Brown, but he had also very carefully read and commented on Federer’s entire text as Federer was writing it. (If you know Federer’s book, you can’t help but be impressed by this.) After graduating from Brown, Bill moved to Princeton where he did his seminal work on varifolds.

So why does image analysis lead rather naturally to geometric measure theory? For the simple reason that edges are a big deal in images, while functions of bounded variation (BV) are a very natural class of functions to use when representing functions with discontinuities (edges). And functions in BV are particularly nice, because they are wild, but not too wild — we can still make sense of derivatives (they are nice measures), and all sorts of other nice properties are still at our disposal. For example, sets, whose characteristic functions are in BV, still have usable, generalized outer normals and as a result, we still have the divergence theorem in such regions. Analysis is still nice, or at least possible. (De Giorgi used this class of functions to solve the minimal surface problem in one of the three papers in 1960 that solved this problem. Another was written by Federer and Fleming, the third by Reifenberg. All three used different methods.)

I have studied, referenced or taught out all the above monographs. I now give more detailed comments on those texts. But I am not going to introduce the subject of geometric measure theory here — that will be the subject of other posts (coming soon), nor will I simply outline the contents of the texts, since I don’t think that adds much value. Instead, I will add the things you can’t get from a perusal of the table of contents.

1. Federer’s 1969 Geometric Measure Theory: To a very large degree, this is still the ultimate go-to reference for the contents of the first 4 (of 5) chapters. This is not to say that that content has not evolved, but rather that it is still the foundation for current work. (For example, Solomon and White have enabled us to avoid the difficult structure theorem in getting existence for minimal surfaces, but that structure theorem is still very important.) This text is also rather notorious for it’s density and difficulty. Some of that difficulty is due to the (sometimes understandable) impatience that readers bring to their reading, but it also seems that the lack of pictures in this particular book is a rather eloquent statement about its accessibility. I have learned that it is much easier to read when you translate nearly everything into pictures (because you can!), though its terseness encourages me to use it as a reference, not a text. But it is a completely indispensable reference! Every students should own a copy and read pieces as needed.
2. Frank Morgan’s Geometric Measure Theory: A Beginner’s Guide: Frank wrote his highly successful text as a path into, and an inspiration for the study of, Federer’s book. In contrast to Federer, Frank draws lots of pictures, many of them very enlightening. Almost everybody in the younger generations — people who could have used Frank’s book first — have first read Frank’s book before using other texts. This is as good a place as any to explain that there have been two main branches of effort in GMT, one focused on variational problems like the minimal surface problem, and the second focused on the geometry of sets and measures, with a particular focus on harmonic measures. Morgan and Federer come out of the first branch, while Mattila’s text, which is commented on below, comes from the second branch. (I will comment below on the two new branches of GMT — GMT in metric spaces and GMT with a view to data analysis.) I always use Frank’s book as an illuminating reference for newcomers. I recommend it very highly as a first exposure and an evangelical tool.
3. Krantz and Parks’ Geometric Integration Theory: At first I was skeptical because I am into nice figures (I am an xfig devotee) and some of Krantz and Parks’ figures were very bad. But then I used the text as a reference when proving the deformation theorem for simplicial complexes (see Simplicial Flat Norm with Scale for the paper) and I was impressed with the exposition. Last year I used it as a text for a graduate class. While there are aspects of the text I still don’t like, I do recommend it as a reference that every student should own and consult. (By the way, Federer originally wanted to name his book Geometric Integration Theory, but didn’t because Whitney had already written a book with the same name. Whitney’s book is relevant for those interested in geometric measure theory, and it is now available from Dover books!)
4. Lin and Yang’s Geometric Measure Theory – an Introduction: I have not used this as a text. I was discouraged from using it by the typos, which it is important to note were also very irritating to Fanghua Lin because he had tried to get the publisher to correct them! But it does contain a very nice selection of topics, maybe the broadest selection in the above texts.
5. Leon Simon’s Lectures on Geometric Measure Theory: As I said above, this is one of my two favorites. I am currently (Fall 2012) teaching a graduate class from this text. The book is not so easy to get, but if you are willing to persist, you can get it from the Centre for Mathematical Analysis at Australian National University. (Though the last time I ordered a bunch of copies, they all had the first page of the index missing and they were clearly photocopy’s of the original print run of the book.) I used Leon’s text one other time, when I taught a short course on GMT at UCLA in the spring of 2007. Why do I like the text so much? There are flaws, like typos and a typewriter type font that takes getting used to and things I would change here and there. But Leon’s selection of topics (not too many!), his versions of theorems, the way in which he gives enough details in proofs, but not too many (leaving many implicit exercises and problems for the reader), and the way in which he puts everything together has generated a book that I really like to study and teach from. I recommend it very highly as a primary text, after or alongside Frank’s book. As a side note, Leon is working on a second edition of this text. It should be very good!
6. Pertti Mattila’s The Geometry of Sets and Measures in Euclidean Spaces: As mentioned above, this text comes from the harmonic analysis branch of the subject. As a result, it does not deal with currents, which were developed for their use in GMT (on minimal surfaces) by Federer and Fleming. Mattila’s book is well written and challenging, but not so challenging that students don’t like it. It has explicit problems, which many students like. (I always preferred implicit, fill-in-the-details or what-if-I-weaken-this-assumption type problems.) It covers lots of material, including Marstrand and Priess’ results (about which I would recommend De Lellis’ notes on Rectifiable Sets, Densities and Tangent Measures ), fractals and connections to singular integrals. It contains things that none of the other books I am commenting on have, and it is the only representative of the harmonic analysis branch of GMT in my selection of books. And it does have a different flavor, as one might expect. I consider it something that every student of GMT should own.
7. Evans and Gariepy’s Measure Theory and Fine Properties of Functions: As noted in my story above, this was the first book I saw on the subject. It deals with the same subjects that the first part of Federer and the first part of Simon deal with. It does not delve into currents. The writing is very clear, the proofs are complete, and the amount of filling in, in the proofs, is consistently small enough to make it fairly fast to study, but often enough to keep you very engaged. I found it inspiring when I first read it and it is still usually the first book I have my students buy. The chapters cover measure theory and integration, Hausdorff measure, Radon measures, area and co-area formulas, Sobolev spaces, BV functions (including detailed development of the structure theorem for sets of finite perimeter), and a final chapter on things like Radamacher’s theoorem and extension theorems like Whitney’s. CRC even lowered the price from 180$to 90$ (it has crept back up a bit) in response to our complaints about the price! As far as prerequisites are concerned, most students find this more accessible after a first course in graduate analysis, but some might be happy with it as an introduction to analysis, as long as some other text, like Royden or Folland is also on hand. I recommend it very highly as a text and reference.
8. Ambrosio, Fusco, and Pallara’s Functions of Bounded Variation and Free Discontinuity Problems: David Caraballo was also the first to tell me about this book. It prepares the reader to deal with the existence results for the Mumford-Shah functional, which is an image analysis functional for used for image segmentation. Ambrosio and De Giorgi proposed the space of special functions of bounded variation (SBV) for use with free boundary problems in 1987. In a 1989 paper by De Giorgi, Carriero, and Leaci, these ideas were used to prove the existence of minimizers for the Mumford-Shah functional. Later Ambrosio developed the theory of SBV more fully and this book is a logical follow-on to these works. I have not used this book, though I have read a small bit here and there. I think it is well liked by students, but it is unreasonably expensive at 250.00 (list price). Shame on Oxford University Press! They should not be following the example of the Dutch profiteers! This is an object lesson in why, if you are a mathematician, you should publish your own book and not give it to some publisher to exploit. OK. Done with my soapbox. If you can afford it, get this book! If you can’t, write to Oxford and complain bitterly about their crazy prices and the fact that they are limiting access to an excellent and fascinating book!
9. Enrico Giusti’s Minimal Surfaces and Functions of Bounded Variation: I read much of Giusti’s book and liked it a great deal. I recommend it as very a good source for the subjects it covers. This book is inspired by De Giorgi’s path to a solution for the minimal surface problem, though it contains more material since it was written over 20 years after that work by De Giorgi. Here is review of Giusti’s book by Fred Almgren. It contains a very detailed account of the contents of the book and some nice history as well. Again, this book is overpriced at 183.00 for the paperback but luckily, Springer (who owns the book’s publisher Birkhauser) sells it for 91.50! So buy it directly from Springer. And I do recommend buying it. You will enjoy studying it if you have any interest in this subject.

Finally, as promised above, a few words about the other two branches of GMT. As mentioned above they are GMT in metric spaces and GMT with a view to data.

As far as I know, GMT in metric spaces had its genesis with Ambrosio and Kirchheim’s paper, Currents in Metric Spaces. It is a very active area of research, including for example, analysis on the Heisenberg group, where paths are allowed tangents in proper subspaces of the tangent space instead allowing the path to have arbitrary directions in the tangent space. (These are called sub-Riemannian spaces.) As far as I can tell, this branch was inspired by the growing area of analysis in metric spaces.

By GMT with a view to data, I mean both GMT applied to data and new GMT inspired by data. That the flow of ideas goes in both direction is very important and the reason why this is a very exciting, productive, high-potential place to work. This is where I am working. It includes the work at the intersection of image analysis and BV/SBV functions, like the work of Rudin, Osher and Fatemi which introduced TV regularization to image analysis and the segmentation functional of Mumford and Shah. Other examples include the work of Jones, Lerman, Schul and Okikiolu on Jones’ beta numbers, the multiscale flat norm work I am doing with collaborators (inspired by the connection between the L1TV functional and the flat norm), as well as the applications of curvature measures (and things like curvature measures) to data. Examples of this last item include the work of Adler, Taylor and Worsley at the intersection of statistics and integral geometry and the work of Chazal, Cohen-Steiner, and Merigot on boundary measures. Many more examples exist, but these examples give a flavor for the kinds of things happening at the intersection of GMT and data.

Actually, the GMT data branch has the other three as subbranches for the simple reason that all three have very useful insights to offer data and because data suggests problems leading to new ideas in each of the three areas.

### Afterword

As mentioned above, I like figures. But, while not having figures in a geometric measure theory text doesn’t make so much sense to me, it is the case that students should be drawing their own figures anyway. This is part of the work involved in making the subject yours, in internalizing the ideas and techniques. One might even argue that books without figures are better for students who must then draw figures for themselves in order to grasp what is going on more fully. But I would not go quite that far. I think that drawing pictures, as many and as often as you can, should be the part of the GMT culture. Figures should be common, and they should also appear in books. I do believe that authors should not try to draw pictures for everything, but should draw just enough to get the students going themselves, help students avoid pitfalls and inspire them as they struggle to master the ideas. At the very least, this discussion helps you see why I can still like Leon Simon’s book so much even though there are no figures in it!

It should also be noted that even though I talk about the two older and the two newer branches of GMT, it would be silly to insist on a bold demarcation of the boundaries between branches and a subsequent classification of everything and everyone. Part of this is because the intersections between branches are very large. Another part of this is because the two newer branches are by their nature agnostic, caring only about generalizable (in the case of metric spaces) or useful (in the case of data analysis) ideas or developments in GMT. Using Federer to supply another example, even though Federer’s most famous paper (the one with Fleming that solved Plateau’s Problem) was focused on calculus of variations, he also established other significant pieces of the foundation for the entire field. It would therefore make no sense at all to try to assign him to a single branch. So thinking about the field as characterized by branches is useful only if you do not take it very seriously, or worse yet, turn the branches into fences impeding travel!