This is the Federer's coarea formula. You find it in this classical but dense book:

Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, New York: Springer-Verlag New York Inc., pp. xiv+676.

This question on derivatives of volumes prompts me with the following curiosity. Let $S_n$ be the surface area of the unit hypersphere in $\mathbb{R}^n$ (avoiding the topologists notation $\mathbb{S}^{n-1}$, for convenience), and treat $S_n$ as a function of a continuous variable $n$ (relying on Euler's $\Gamma$-function). Now, setting the derivative $\frac{dS_n}{dn}=0$ leads to $n=735\dots$; hence the "amusing" fact that the $7$-dimensional unit hypersphere has the maximum surface area.

On the other hand, we have some striking results such as the first among the spheres having exotic structures is the $7$-dim sphere (Milnor). Plus, while no even dimensional spheres are parallelizable, there are only $1$-d, $3$-d and $7$-d among the odds. Then it stops.

My question is: what is so special about the $7$-dimensional sphere? I wonder.

This is Federer's coarea formula. You find it in this classical but dense book:

Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, New York: Springer-Verlag New York Inc., pp. xiv+676.

This question on derivatives of volumes prompts me with the following curiosity. Let $S_n$ be the surface area of the unit hypersphere in $\mathbb{R}^n$ (avoiding the topologists notation $\mathbb{S}^{n-1}$, for convenience), and treat $S_n$ as a function of a continuous variable $n$ (relying on Euler's $\Gamma$-function). Now, setting the derivative $\frac{dS_n}{dn}=0$ leads to $n=7.35\dots$; hence the "amusing" fact that the $7$-dimensional unit hypersphere has the maximum surface area.

On the other hand, we have some striking results such as the first among the spheres having exotic structures is the $7$-dim sphere (Milnor). Plus, while no even dimensional spheres are parallelizable, there are only $1$-d, $3$-d and $7$-d among the odds. Then it stops.

My question is: what is so special about the $7$-dimensional sphere? I wonder.

This is the Federer's coarea formula. You find it in this classical but dense book.

Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, New York: Springer-Verlag New York Inc., pp. xiv+676

This is the Federer's coarea formula. You find it in this classical but dense book:

Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, New York: Springer-Verlag New York Inc., pp. xiv+676.

This question on derivatives of volumes prompts me with the following curiosity. Let $S_n$ be the surface area of the unit hypersphere in $\mathbb{R}^n$ (avoiding the topologists notation $\mathbb{S}^{n-1}$, for convenience), and treat $S_n$ as a function of a continuous variable $n$ (relying on Euler's $\Gamma$-function). Now, setting the derivative $\frac{dS_n}{dn}=0$ leads to $n=735\dots$; hence the "amusing" fact that the $7$-dimensional unit hypersphere has the maximum surface area.

On the other hand, we have some striking results such as the first among the spheres having exotic structures is the $7$-dim sphere (Milnor). Plus, while no even dimensional spheres are parallelizable, there are only $1$-d, $3$-d and $7$-d among the odds. Then it stops.

My question is: what is so special about the $7$-dimensional sphere? I wonder.