Today I encountered a problem concerning the Hausdorff measure on the sphere.

Let $n\ge 3$ and $\Bbb S^{n-1}$ denote the $n-1$-dimensional unit sphere in $\Bbb R^{n}$ namely, $$\Bbb S^{n-1}= \{x \in \Bbb R^{n}: \|x\|=1\}$$

For $v\in \Bbb S^{n-1}$ fixed we define $$\Bbb S_v^{n-2}=\Bbb S^{n-1}\cap (\Bbb R v)^\bot$$ where $(\Bbb R v)^\bot$ is the orthogonal of $v$ in $\Bbb R^{n}$.

Question: How can one prove the $n-2$-Hausdorff of $\Bbb S_v^{n-2}$ equals the surface measure of $\Bbb S^{n-2}$ (which is the $n-2$-dimensional unit sphere in $\Bbb R^{n-1}$ ) that $$\mathcal{H}^{n-2}(\Bbb S_v^{n-2})=|\Bbb S^{n-2}|$$

This may not help but I would like to remind the following formula: $$ |\Bbb S^{n-1}| =\frac{2\pi^{n/2}}{\Gamma(n/2)}. $$

Definition: Given $E\subset \Bbb R^n$ and $s\in \Bbb R_+$ by definition the $s$-dimensional Hausdorff measure of $E$ is, $$\mathcal{H}^{s}(E)=\lim_{\delta\to 0^+}\mathcal{H}_\delta^{s}(E)$$ with $$\mathcal{H}_\delta^{s}(E)=\inf\left\{\sum_{i=0}^{\infty}\left(diam(U_i)\right)^s:~~ E\subset\bigcup_{i=0}^{\infty}U_i, ~~~ diam(U_i)\le \delta\right\}$$ where $(U_i)_{i}$ is an open covering of $E$ and $diam(U_i)=\sup\left\{d(x,y):~~ x,y\in U_i\right\}$

  • 2
    $\begingroup$ Depends on how "the surface measure" is defined and how much is established about it. For example, it's the only rotation-invariant measure up to normalization. Since $\mathcal H^s$ is also rotation invariant, the problem boils down to finding the constant of proportionality. Alternatively (not relying on the special structure of the sphere) one can prove change-of-variable formulas for each measure, describing how they are pushed by diffeomorphisms. Pushing both measures by coordinate maps, this reduces the problem to a domain in $\mathbb{R}^2$. $\endgroup$ – user357151 Dec 28 '17 at 19:33

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