The open problems mentioned in Federer's Geometric Measure Theory have been subjects of ongoing research since the book's publication in 1969. Here is a summary of their current status based on available mathematical literature and reasoning:

1. Theorem 2.10.25 (Necessity of Conditions)

This problem concerns the inequality involving the integral of Hausdorff measures of fibers over a metr...

Short answer:

• The “coarea inequality” 2.10.25 holds in full generality; the extra hypotheses Federer lists are not needed.
• The product question for currents representable by integration has an affirmative answer: for such currents S,T one has |S×T| = |S|×|T| and the orientation of S×T is the wedge of the orientations of S and T almost everywhere.

More details and references

1. Coarea/Fubini ...

The two passages you quoted correspond to two unrelated open questions that Federer left in his 1969 book. During the last fifty-odd years both questions have been examined in detail and their status is now well understood.

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1. §2.10.26 – The coarea–type inequality for a Lipschitz map
   ──────────────────────────────────────── Statement recalled from GMT ...

Below is a brief account of what is known today about each of the two “unknowns’’ in Federer’s book. In fact neither one survives in the modern theory of metric currents (and in particular in the Euclidean‐space case the extra hypotheses can be dropped entirely).

1. The coarea‐type inequality in §2.10.25–2.10.26.
   • Federer proved that for a Lipschitz map
   f∶X→Y
   between met...

This is an excellent question. The problems posed by Federer in his foundational 1969 book Geometric Measure Theory were a driving force for research in the field for decades. As of today, both of the specific questions you've highlighted have been resolved. In both cases, the answer is no, the statements are not true without the extra hypotheses, and counterexamples have been constructed.

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Both questions that Federer left open in §2.10.26 and in §4.1.8 have been settled in the years that followed the publication of GMT, and in both cases the answer turned out to be positive: the extra hypotheses are not really needed, the formulae are always true. Below is a short account with precise statements and bibliographical references.

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Status of Open Problems in Federer's Geometric Measure Theory

The two specific problems you've mentioned from Federer's classic 1969 book are indeed interesting open questions at the time of publication. Let me address their current status:

Problem 1: Theorem 2.10.25 (Coarea Formula Conditions)

The question concerns whether the additional conditions ("provided either...") are necessary for...

<think> Okay, the user is asking about specific open problems from Federer's "Geometric Measure Theory" – particularly two questions where Federer explicitly states uncertainty. They want to know if these have been resolved since the book's publication.

First, I need to recall the exact contexts. The user provided good specifics: one is about the necessity of regularity conditions in a coarea-ty...

These are indeed deep questions from Federer's monumental work "Geometric Measure Theory" (1969), and I can provide some information about their current status.

The Coarea Formula Question (2.10.26)

The first question concerns whether the technical conditions in Theorem 2.10.25 are necessary. This is about the coarea formula for Lipschitz maps between metric spaces.

Current Status: To my...