Open problems in Federer's Geometric Measure Theory

Question

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this equations are always true" p.361 4.1.8, or "I do not know..." p.189 2.10.26. Are there counterexamples or proofs of these asumptions? I add some details : in theorem 2.10.25, if $f:X\rightarrow Y$ is a lipschitzian map of metric spaces, $A\subset X$, $0\leq k<\infty$, and $0\leq m<\infty$, then $$ \int_Y^*\mathscr{H}^k(A\cap f^{-1}\{y\})d\mathscr{H}^m(y)\leq (\mathrm{Lip} f)^m \frac{\alpha(k)\alpha(m)}{\alpha(k+m)}\mathscr{H}^{k+m}(A). $$ (where $\mathscr{H}^n$ is the Hausdorff measure associated to the metrics on $X$ and $Y$, and $\int^*$ is the upper integral) provided either $\{y\in Y, \mathscr{H}^k(A\cap f^{-1})>0\}$ is the union of countable family of sets with finite $\mathscr{H}^m$ measure, or $Y$ is boundedly compact (each close bounded subset is compact). The question Federer asks is to determine if everything after "provided either..." is necessary.

The other question is about currents : let $S$, $T$ be two currents on open subsets $A$,$B$ of euclidean spaces, of degree $i$ and $j$ : if $S$, $T$ are representable by integration, do we always have $\Vert S\times T\Vert=\Vert S\Vert\times\Vert T\Vert$, and $\overrightarrow{S\times T}(a,b)=(\wedge_i p)\overrightarrow{S}(a)\wedge(\wedge_j q)\overrightarrow{T}(b)$ for $\Vert S\times T\Vert$ almost all $(a,b)\in A\times B$, where $p:A\rightarrow A\times B$ and $q:B\rightarrow A\times B$ are the canonical injections (it is indeed the case if either $\overrightarrow{S}$ or $\overrightarrow{T}$ is simple $\Vert S\Vert\times\Vert T\Vert$ almost everywhere).

Edit : I found the answer of the first question in a book of Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, where there is a mention of a more general form for abrbitrary separable metric spaces ($X$ and $Y$ are assumed to be separable). This result was proved by Roy O. Davies, in the article Increasing Sequences of Sets and Hausdorff Measure (Proc. London Math. Soc. 20, 222-236, 1970).

Answer 1

For a full proof of the coarea inequality in the context of metric spaces using the results of Davies you may also be interested in the PhD thesis of L. Reichel. See Theorem 7.1 in

http://e-collection.library.ethz.ch/eserv/eth:289/eth-289-02.pdf

Regarding currents of low dimension or codimension, a proof of the second question was obtained by F. Morgan (The exterior algebra ΛkRn and area minimization, Linear Algebra and its Applications Volume 66, April 1985, Pages 1–28). It is also stated in there that the general case is still open. I don't know if this has changed since then...

Answer 2

Theorem. The following inequality is true for any metric spaces $X,Y$, any $A\subset X$, any Lipschitz map $f:X\to Y$ and any real numbers $k,m\geq 0$. $$ \int_Y^*\mathscr{H}^k(A\cap f^{-1}\{y\})d\mathscr{H}^m(y)\leq (\mathrm{Lip} f)^m \frac{\alpha(k)\alpha(m)}{\alpha(k+m)}\mathscr{H}^{k+m}(A). $$

Fereder could prove the inequality in cases when he could establish equality $\lim_{\delta\to 0^+}\lambda_\delta(g)=\int^*g\, d\psi$ (see page 187 in Federer's book [F1]). For example he could prove Theorem when $Y$ is boundedly compact i.e. bounded and closed sets in $Y$ are compact (in fact Federer's result was originally proved in [F2]). In general he could prove the inequality $$ \lim_{\delta\to 0^+}\lambda_\delta(g)\leq\int^*g\, d\psi. $$ Then he asked [F1, page 187]:

The general problem whether or not the preceding inequality can always be replaced by the corresponding equation is unsolved.

A positive answer to this question would imply Theorem in the general case. The problem was answered in the positive by Davies [D, page 236]:

Note added 8 September 1969. H. Federer tells me that this work answers a question he raised in Geometric measure theory (Berlin, 1969) [...]

However, there is no proof of Theorem in Davies' paper and one could get the proof only by reading Federer's proof, by reading Davies' paper and by understanding how to combine the two together.

Surprisingly, it wasn’t until 2009 when Reichel [R] in his PhD thesis, re-wrote a complete proof of Theorem in its full generality, by following the original proof of Federer while making use of Davies’ result. Until recently Reichel’s thesis was the only place with a complete proof of Theorem, except that Reichel did not include the proof of Davies’ theorem. Davies’ theorem is however, very difficult.

You can find a new and a complete proof of Theorem that avoids Davis' result in:

https://arxiv.org/abs/2006.00419

This paper also contains a detailed history of the problem.

[D] Davies, R. O.: Increasing sequences of sets and Hausdorff measure, Proc. London Math. Soc. 20 (1970), 222-236.

[F1] Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969.

[F2] Federer, H.: Some integralgeometric theorems. Trans. Amer. Math. Soc. 77 (1954), 238–261.

[R] Reichel, L. P.: The coarea formula for metric space valued maps. Ph.D. thesis, ETH Zurich, 2009.