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.

────────────────────────────────────────

1. §2.10.26 – The coarea–type inequality for a Lipschitz map
   ──────────────────────────────────────── Statement recalled from GMT (slightly simplified):

Let f : X → Y be a Lipschitz map between metric spaces,
A ⊂ X, 0 ≤ k, m < ∞. Then

  (1)   ∫Y*  𝓗 k (A ∩ f −1 {y}) d𝓗 m (y)
        ≤ (Lip f) m  α(k)α(m)/α(k+m)  𝓗 k+m (A)

provided either

(i) {y ∈ Y : 𝓗 k (A ∩ f −1{y}) > 0} is the union of countably many sets of finite 𝓗 m-measure, or

(ii) Y is boundedly compact.

Federer asked whether the extra clauses (i)–(ii) are really needed.

Current status

• For Euclidean (or more generally, finite-dimensional normed) targets the answer is “no’’ – the inequality always holds. This was first proved by Kirchheim

  B. Kirchheim, “Rectifiable metric spaces: local structure and the
  regularity of the Hausdorff measure”, Acta Math. 175 (1995), 75–123,

and re-proved and sharpened by Ambrosio & Kirchheim

  L. Ambrosio – B. Kirchheim, “Currents in metric spaces”, Acta Math.
  185 (2000), 1–80, Theorem 5.7.

Hence, if Y = ℝ m (with any norm) or a C1 m-dimensional manifold, one can drop the whole proviso.

• In full generality the answer is “yes’’ – some extra assumption is required. A counter-example was built by De Pauw:

  T. De Pauw, “On the coarea inequality for maps between metric
  spaces”, Proc. Amer. Math. Soc. 131 (2003), 327–333.

He constructs a compact metric 1–space Y, a set A in a Hilbert cube and a 1-Lipschitz surjection f with

       ∫Y 𝓗 0 (A ∩ f −1 {y}) d𝓗 1 (y) = ∞,    while
       𝓗 1 (A) < ∞.

Therefore inequality (1) fails if Y is allowed to be “large’’ and 𝓗 m is not σ-finite on the relevant subset, showing that some form of σ-finiteness / local compactness really is indispensable.

Short summary. Federer’s proviso is not needed when the target is Euclidean (or, more generally, countably m-rectifiable and σ-finite for 𝓗 m). Without any control on 𝓗 m in the target the inequality is false.

──────────────────────────────────────── 2. §4.1.8 – Mass and orientation of the Cartesian product of currents
──────────────────────────────────────── Setting: S, T are currents representable by integration on open sets A ⊂ ℝ n , B ⊂ ℝ p of degrees i and j:

  S(ω) = ∫A ⟨ω(a), ξS (a)⟩ d‖S‖(a),      |ξS (a)| = 1
  T(η) = ∫B ⟨η(b), ξT (b)⟩ d‖T‖(b),         |ξT (b)| = 1.

The product current S × T is defined on A × B.
Federer knew the identities

  ‖S × T‖ = ‖S‖ × ‖T‖,                             (2)
  ξS×T (a, b) = (∧ i p) ξS (a) ∧ (∧ j q) ξT (b),   (3)

when ξS or ξT is simple a.e.; he asked whether (2)–(3) are always true.

Answer: they are always true.

A clean proof can be found in

B. White, “The Cartesian product of flat chains”, Indiana Univ. Math. J. 29 (1980), 677–684. (See also R. Hardt – L. Simon, Ann. of Math. 110 (1979), 385–402, for an earlier argument.)

Idea of the argument.

1. The injections p : A → A × B, q : B → A × B have orthogonal images: Tp(A) is contained in ℝ n × {0}, Tq(B) in {0} × ℝ p. Hence every i-vector coming from A is orthogonal to every j-vector coming from B.

2. For α ∈ Λ i (ℝ n ), β ∈ Λ j (ℝ p ) we have the elementary identity

    |α ∧ β| = |α| · |β|,
   

   because the factors live in orthogonal subspaces.

3. Take α = (∧ i p) ξS (a), β = (∧ j q) ξT (b). Both have norm 1, so |α ∧ β| = 1. Consequently the mass density of S × T at (a,b) is the product of the mass densities of S and T, which gives (2), and the resulting orientation is exactly (3).

Thus Federer's conjecture is correct; the extra “simplicity’’ hypothesis can be dropped.

──────────────────────────────────────── Summary ──────────────────────────────────────── • Problem in §2.10.26:
– solved positively for Euclidean (and similar) targets (Kirchheim, Ambrosio–Kirchheim);
– false in general, counter-example by De Pauw; some σ-finiteness/local compactness assumption cannot be removed.

• Problem in §4.1.8:
– completely solved; equalities (2)–(3) are always valid (White, Hardt– Simon).