# Open problems in Federer's Geometric Measure Theory

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).

• Welcome to the site! It could help attract more interest to the question if it were possible to understand it without having the book handy. Perhaps you could reproduce the things you ask about here. You can expand the question by clicking "edit" just below it. – user9072 Jul 19 '14 at 13:00
• Thank you quid, I've just added some extra informations. – Paul-Benjamin Jul 19 '14 at 13:49

## 1 Answer

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...

• Thank you very much rozu, I was unaware of this PhD thesis, so you've answered to more than I expected, I am truly grateful! – Paul-Benjamin Dec 20 '14 at 9:32