I have a very indirect proof of the following property involving a parametrized integral. If $a,a_1,\ldots,a_n\in\mathbb R^n$ (here $n\ge2$), let me denote $V(a,a_1,\ldots,a_n)$ the volume of the simplex spanned by these $n+1$ points. Then $$\sup_{a_1,\ldots,a_n}\int_{\mathbb R^n}\left[\frac{V(x,a_1,\ldots,a_n)^2}{\left(\prod_1^n|x-a_i|\right)^{n+1}}\right]^{\frac1{n-1}}dx<+\infty.$$ Remark that the integral, $I(a_1,\ldots,a_n)$, is homogeneous of degree zero in the argument $(a_1,\ldots,a_n)$.

Mind also that it can be bounded by $$\int_{\mathbb R^n}\frac{dx}{\prod_1^n|x-a_i|},$$ but the latter integral diverges ; and its restriction to a ball $B(0;R)$, while finite, tends to infinity as all $a_j$'s converge to $0$.

Is there an elementary proof of this boundedness, or is it notoriously difficult ? If elementary, do you have an explicit reference ?

Remark that if $n=2$, then $I(a_1,a_2)$ is a constant, because it is invariant under isometries and homogeneous of degree $0$, thus $\equiv I(0,\vec e_1)$.

I have a very indirect proof of the following property involving a parametrized integral. If $a,a_1,\ldots,a_n\in\mathbb R^n$ (here $n\ge2$), let me denote $V(a,a_1,\ldots,a_n)$ the volume of the simplex spanned by these $n+1$ points. Then $$\sup_{a_1,\ldots,a_n}\int_{\mathbb R^n}\left[\frac{V(x,a_1,\ldots,a_n)^2}{\left(\prod_1^n|x-a_i|\right)^{n+1}}\right]^{\frac1{n-1}}dx<+\infty.$$ Remark that the integral, $I(a_1,\ldots,a_n)$, is homogeneous of degree zero in the argument $(a_1,\ldots,a_n)$.

Mind also that the naive bound $$\int_{\mathbb R^n}\frac{dx}{\prod_1^n|x-a_i|}$$ diverges ; and the restriction of the latter integral to a ball $B(0;R)$, while finite, tends to infinity as all $a_j$'s converge to $0$.

Is there an elementary proof of the boundedness of $(a_1,\ldots,a_n)\mapsto I$, or is it notoriously difficult ? If elementary, do you have an explicit reference ?

Remark that if $n=2$, then $I(a_1,a_2)$ is a constant, because it is invariant under isometries and homogeneous of degree $0$, thus $\equiv I(0,\vec e_1)$.

I have a very indirect proof of the following property involving a parametrized integral. If $a,a_1,\ldots,a_n\in\mathbb R^n$, let me denote $V(a,a_1,\ldots,a_n)$ the volume of the simplex spanned by these $n+1$ points. Then $$\sup_{a_1,\ldots,a_n}\int_{\mathbb R^n}\left[\frac{V(x,a_1,\ldots,a_n)^2}{\left(\prod_1^n|x-a_i|\right)^{n+1}}\right]^{\frac1{n-1}}dx<+\infty.$$ Remark that the integral is homogeneous of degree zero in the argument $(a_1,\ldots,a_n)$.

Mind also that it can be bounded by $$\int_{\mathbb R^n}\frac{dx}{\prod_1^n|x-a_i|},$$ but the latter integral diverges ; and its restriction to a ball $B(0;R)$, while finite, tends to infinity as all $a_j$'s converge to $0$.

Is there an elementary proof of this boundedness, or is it notoriously difficult ? If elementary, do you have an explicit reference ?

I have a very indirect proof of the following property involving a parametrized integral. If $a,a_1,\ldots,a_n\in\mathbb R^n$ (here $n\ge2$), let me denote $V(a,a_1,\ldots,a_n)$ the volume of the simplex spanned by these $n+1$ points. Then $$\sup_{a_1,\ldots,a_n}\int_{\mathbb R^n}\left[\frac{V(x,a_1,\ldots,a_n)^2}{\left(\prod_1^n|x-a_i|\right)^{n+1}}\right]^{\frac1{n-1}}dx<+\infty.$$ Remark that the integral, $I(a_1,\ldots,a_n)$, is homogeneous of degree zero in the argument $(a_1,\ldots,a_n)$.

Mind also that it can be bounded by $$\int_{\mathbb R^n}\frac{dx}{\prod_1^n|x-a_i|},$$ but the latter integral diverges ; and its restriction to a ball $B(0;R)$, while finite, tends to infinity as all $a_j$'s converge to $0$.

Is there an elementary proof of this boundedness, or is it notoriously difficult ? If elementary, do you have an explicit reference ?

Remark that if $n=2$, then $I(a_1,a_2)$ is a constant, because it is invariant under isometries and homogeneous of degree $0$, thus $\equiv I(0,\vec e_1)$.