Mean width of compact shape in 2D
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Mean width of a convex subset $Kinmathbb{R}^n$ is defined by
$$W(K)=dfrac{1}{sigma_n}int_{mathbb{S}^{n-1}}mathcal{L}(P_u(K))du,$$
where $mathbb{S}^{n-1}$ is the unit sphere in $mathbb{R}^n$, $sigma_n$ is the volume of $mathbb{S}^{n-1}$ and $mathcal{L}(P_u(K))$ is length of the orthogonal projection of $K$ on the 1-dimensional space $mathbb{R}u$ generated by u.
On Wikipedia, there is a remark for the case of 2-dimension that the mean width of any compact shape $S$ in two dimensions is $p/pi$, where $p$ is the perimeter of the convex hull of $S$. Can you give me the references such as books or papers for this remark (2D case)?
About mean width on Wikipedia: https://en.wikipedia.org/wiki/Mean_width
geometry analysis convex-analysis
asked Dec 18 '18 at 2:47
Fuid NirantoFuid Niranto
12
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add a comment |
$begingroup$
Mean width of a convex subset $Kinmathbb{R}^n$ is defined by
$$W(K)=dfrac{1}{sigma_n}int_{mathbb{S}^{n-1}}mathcal{L}(P_u(K))du,$$
where $mathbb{S}^{n-1}$ is the unit sphere in $mathbb{R}^n$, $sigma_n$ is the volume of $mathbb{S}^{n-1}$ and $mathcal{L}(P_u(K))$ is length of the orthogonal projection of $K$ on the 1-dimensional space $mathbb{R}u$ generated by u.
On Wikipedia, there is a remark for the case of 2-dimension that the mean width of any compact shape $S$ in two dimensions is $p/pi$, where $p$ is the perimeter of the convex hull of $S$. Can you give me the references such as books or papers for this remark (2D case)?
About mean width on Wikipedia: https://en.wikipedia.org/wiki/Mean_width
geometry analysis convex-analysis
asked Dec 18 '18 at 2:47
Fuid NirantoFuid Niranto
12
$endgroup$
$begingroup$
I don't know about references to books or papers, but the proof itself is not too hard, see e.g. math.stackexchange.com/a/93743/856
$endgroup$
– Rahul
Dec 18 '18 at 3:12
add a comment |
$begingroup$
Mean width of a convex subset $Kinmathbb{R}^n$ is defined by
$$W(K)=dfrac{1}{sigma_n}int_{mathbb{S}^{n-1}}mathcal{L}(P_u(K))du,$$
where $mathbb{S}^{n-1}$ is the unit sphere in $mathbb{R}^n$, $sigma_n$ is the volume of $mathbb{S}^{n-1}$ and $mathcal{L}(P_u(K))$ is length of the orthogonal projection of $K$ on the 1-dimensional space $mathbb{R}u$ generated by u.
On Wikipedia, there is a remark for the case of 2-dimension that the mean width of any compact shape $S$ in two dimensions is $p/pi$, where $p$ is the perimeter of the convex hull of $S$. Can you give me the references such as books or papers for this remark (2D case)?
About mean width on Wikipedia: https://en.wikipedia.org/wiki/Mean_width
geometry analysis convex-analysis
asked Dec 18 '18 at 2:47
Fuid NirantoFuid Niranto
12
$endgroup$
Mean width of a convex subset $Kinmathbb{R}^n$ is defined by
$$W(K)=dfrac{1}{sigma_n}int_{mathbb{S}^{n-1}}mathcal{L}(P_u(K))du,$$
where $mathbb{S}^{n-1}$ is the unit sphere in $mathbb{R}^n$, $sigma_n$ is the volume of $mathbb{S}^{n-1}$ and $mathcal{L}(P_u(K))$ is length of the orthogonal projection of $K$ on the 1-dimensional space $mathbb{R}u$ generated by u.
On Wikipedia, there is a remark for the case of 2-dimension that the mean width of any compact shape $S$ in two dimensions is $p/pi$, where $p$ is the perimeter of the convex hull of $S$. Can you give me the references such as books or papers for this remark (2D case)?
About mean width on Wikipedia: https://en.wikipedia.org/wiki/Mean_width
geometry analysis convex-analysis
geometry analysis convex-analysis
asked Dec 18 '18 at 2:47
Fuid NirantoFuid Niranto
12
asked Dec 18 '18 at 2:47
Fuid NirantoFuid Niranto
12
edited Jan 3 at 14:33
Fuid Niranto
asked Dec 18 '18 at 2:47
Fuid NirantoFuid Niranto
12
asked Dec 18 '18 at 2:47
Fuid NirantoFuid Niranto
12
asked Dec 18 '18 at 2:47
Fuid NirantoFuid Niranto
12
12
$begingroup$
I don't know about references to books or papers, but the proof itself is not too hard, see e.g. math.stackexchange.com/a/93743/856
$endgroup$
– Rahul
Dec 18 '18 at 3:12
add a comment |
$begingroup$
I don't know about references to books or papers, but the proof itself is not too hard, see e.g. math.stackexchange.com/a/93743/856
$endgroup$
– Rahul
Dec 18 '18 at 3:12
$begingroup$
I don't know about references to books or papers, but the proof itself is not too hard, see e.g. math.stackexchange.com/a/93743/856
$endgroup$
– Rahul
Dec 18 '18 at 3:12
$begingroup$
I don't know about references to books or papers, but the proof itself is not too hard, see e.g. math.stackexchange.com/a/93743/856
$endgroup$
– Rahul
Dec 18 '18 at 3:12
add a comment |
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$begingroup$
I don't know about references to books or papers, but the proof itself is not too hard, see e.g. math.stackexchange.com/a/93743/856
$endgroup$
– Rahul
Dec 18 '18 at 3:12