Chapter 29

The Area FormulaI am grateful to those who have found errors in this material, some of which were egregious.I would not have found these mistakes because I never teach this material and I don’t use itin my research. I do think it is wonderful mathematics however.

To begin with is a simple theorem about extending Lipschitz functions.

Theorem 29.0.1 If h : Ω→ Rm is Lipschitz, then there exists h : Rp→ Rm which extendsh and is also Lipschitz.

Proof: It suffices to assume m = 1 because if this is shown, it may be applied to thecomponents of h to get the desired result. Suppose

|h(x)−h(y)| ≤ K |x−y|. (29.0.1)

Defineh(x)≡ inf{h(w)+K |x−w| : w ∈Ω}. (29.0.2)

If x ∈Ω, then for all w ∈Ω,

h(w)+K |x−w| ≥ h(x)

by 29.0.1. This shows h(x)≤ h(x). But also you could take w = x in 29.0.2 which yieldsh(x)≤ h(x). Therefore h(x) = h(x) if x ∈Ω.

Now suppose x,y ∈ Rp and consider∣∣h(x)−h(y)

∣∣. Without loss of generality assumeh(x)≥ h(y) . (If not, repeat the following argument with x and y interchanged.) Pick w∈Ω

such thath(w)+K |y−w|− ε < h(y).

Then ∣∣h(x)−h(y)∣∣= h(x)−h(y)≤ h(w)+K |x−w|−

[h(w)+K |y−w|− ε]≤ K |x−y|+ ε.

Since ε is arbitrary, ∣∣h(x)−h(y)∣∣≤ K |x−y|

29.1 Estimates for Hausdorff MeasureIt was shown in Lemma 28.4.5 that

H n(FA) = det(U)mn(A)

where F = RU with R preserving distances and U a symmetric matrix having all positiveeigenvalues. The area formula gives a generalization of this simple relationship to the casewhere F is replaced by a nonlinear mapping h. It contains as a special case the earlierchange of variables formula. There are two parts to this development. The first part isto generalize Lemma 28.4.5 to the case of nonlinear maps. When this is done, the areaformula can be presented.

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