Return to Question

According to my calculations, the average height of the unit hyper-hemisphere in $\mathbb{R}^n$ is given by the formula

$$h_n:=\frac{S_{n-2}}{V_{n-2}} \int_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$$

where $S_{n-2}$, $V_{n-2}$ is respectively the surface and volume of the hypersphere in $\mathbb R^{n-1}$.

Question: How can we understand the quantity $h_n$? In particular, is $h_n$ increasing? Does it tend to $1$ as $n$ goes to infinity?

Edit: The formula i quote above is not the average height of unit hemisphere.

Consider the quantity

$$h_n:=\frac{S_{n-2}}{V_{n-2}} \int_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$$

where $S_{n-2}$, $V_{n-2}$ is respectively the surface and volume of the hypersphere in $\mathbb R^{n-1}$.

Question: How can we understand $h_n$? In particular, is $h_n$ increasing? Does it tend to $1$ as $n$ goes to infinity?

PS: The formula i quote above is not the average height of unit hemisphere. By average height of unit hemisphere i mean the integral $\int_{0}^1 |x_1| d \mu$, where $\mu$ is the uniform measure on $S^{D-1}$ and $x=(x_1,\dots,x_D)$ is the coordinate representation of a point $x \in S^{D-1}$. This integral i have calculated to be $\frac{(D-1)!!}{(D-2)!!} \frac{2}{\pi}$ if $D$ is even, and $\frac{(D-1)!!}{(D-2)!!}$ if $D$ is odd.

According to my calculations, the average height of the unit hyper-hemisphere in $\mathbb{R}^n$ is given by the formula

$$h_n:=\frac{S_{n-2}}{V_{n-2}} \int_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$$

where $S_{n-2}$, $V_{n-2}$ is respectively the surface and volume of the hypersphere in $\mathbb R^{n-1}$.

Question: How can we understand the quantity $h_n$? In particular, is $h_n$ increasing? Does it tend to $1$ as $n$ goes to infinity?

According to my calculations, the average height of the unit hyper-hemisphere in $\mathbb{R}^n$ is given by the formula

$$h_n:=\frac{S_{n-2}}{V_{n-2}} \int_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$$

where $S_{n-2}$, $V_{n-2}$ is respectively the surface and volume of the hypersphere in $\mathbb R^{n-1}$.

Question: How can we understand the quantity $h_n$? In particular, is $h_n$ increasing? Does it tend to $1$ as $n$ goes to infinity?

Edit: The formula i quote above is not the average height of unit hemisphere.

According to my calculations, the average height of the unit hyper-hemisphere in $\mathbb{R}^n$ is given by the formula

$h_n:=\frac{S_{n-2}}{V_{n-2}} \int_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$,

where $S_{n-2}$, $V_{n-2}$ is respectively the surface and volume of the hypersphere in $\mathbb{R}^{n-1}$.

Question: How can we understand the quantity $h_n$? In particular, is $h_n$ increasing? Does it tend to $1$ as $n$ goes to infinity?

According to my calculations, the average height of the unit hyper-hemisphere in $\mathbb{R}^n$ is given by the formula

$$h_n:=\frac{S_{n-2}}{V_{n-2}} \int_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$$

where $S_{n-2}$, $V_{n-2}$ is respectively the surface and volume of the hypersphere in $\mathbb R^{n-1}$.

Question: How can we understand the quantity $h_n$? In particular, is $h_n$ increasing? Does it tend to $1$ as $n$ goes to infinity?