# 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!