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LayZ
LayZ
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Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral: $$ \int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx. $$ Since the integration is taking over the sphere, we have rotation invariance and the value of the integration only depends on the value of $\langle v,w \rangle$. Now the question is to show that the integral is a non-decreasing function w.r.t $\langle v,w \rangle$, for $\langle v,w \rangle > 0$. I believe that this question is connected to the problem of packing two pairs of two antipodal points on a sphere (i.e. four diametrically symmetric points), but I could not show the connection. Any help is much appreciated.

Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral: $$ \int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx. $$ Since the integration is taking over the sphere, we have rotation invariance and the value of the integration only depends on the value of $\langle v,w \rangle$. Now the question is to show that the integral is a non-decreasing function w.r.t $\langle v,w \rangle$, for $\langle v,w \rangle > 0$. I believe that this question is connected to the problem of packing two points on a sphere, but I could not show the connection. Any help is much appreciated.

Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral: $$ \int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx. $$ Since the integration is taking over the sphere, we have rotation invariance and the value of the integration only depends on the value of $\langle v,w \rangle$. Now the question is to show that the integral is a non-decreasing function w.r.t $\langle v,w \rangle$, for $\langle v,w \rangle > 0$. I believe that this question is connected to the problem of packing two pairs of two antipodal points on a sphere (i.e. four diametrically symmetric points), but I could not show the connection. Any help is much appreciated.

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Carlo Beenakker
Carlo Beenakker
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How the value Monotonic dependence on an angle of an integral changes as we implicitly changeover the parameters inside it$n$-sphere

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LayZ
LayZ
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How the value of an integral changes as we implicitly change the parameters inside it

Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral: $$ \int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx. $$ Since the integration is taking over the sphere, we have rotation invariance and the value of the integration only depends on the value of $\langle v,w \rangle$. Now the question is to show that the integral is a non-decreasing function w.r.t $\langle v,w \rangle$, for $\langle v,w \rangle > 0$. I believe that this question is connected to the problem of packing two points on a sphere, but I could not show the connection. Any help is much appreciated.