Let $n\in \mathbb N\setminus\{0,1\}$, $E=\{\xi\in \mathbb R^{2n}\mid \xi_1=0\}$, $K\subset E$ be a $(2n-1)$-dimensional convex polyhedral cone with apex at the origin and $B\subset\mathbb R^{2n}$ the unit ball about the origin. If ${\rm vol}_{2n-1}(K\cap B)$ denotes the Lebesgue measure of $K\cap B$, I would like to know if the following equality is correct:

\begin{eqnarray} \int_{K\cap B} \frac{\Vert \xi\Vert^2+\xi_2^2}{\Vert \xi\Vert^{n+1}}\,d\xi_2\wedge\ldots\wedge d\xi_{2n} =2{\rm vol}_{2n-1}(K\cap B)\,. \end{eqnarray}

The preceding formula is a consequence of some relations about Kazarnovskii mixed pseudovolume (a generalization of Minkowski mixed volume) but it seem to me it can hardly be true, at least in this generality.

My attempt to prove it uses spherical coordinates. Setting $d\vartheta=d\vartheta_1\wedge\ldots\wedge d\vartheta_{2n-2}$ and passing to polar coordinates in $E$ yields the equivalent relation: \begin{eqnarray} \int_{K\cap B} \rho^{n-1}[1+(\cos\vartheta_1)^2]\prod_{\ell=1}^{2n-3}(\sin\vartheta_\ell)^{2n-2-\ell} d\rho\wedge d\vartheta = 2\int_{K\cap B} \rho^{2n-2}\prod_{\ell=1}^{2n-3}(\sin\vartheta_\ell)^{2n-2-\ell} d\rho\wedge d\vartheta\,, \end{eqnarray} or, equivalently, \begin{eqnarray} \int_{K\cap\partial B} \left[\frac{2-(\sin\vartheta_1)^2}{n}-\frac{2}{2n-1}\right](\sin\vartheta_1)^{2n-3}\prod_{\ell=2}^{2n-3}(\sin\vartheta_\ell)^{2n-2-\ell}d\vartheta = 0\,.\label{integrale-lungo} \end{eqnarray} Since \begin{eqnarray} \left[\frac{2-(\sin\vartheta_1)^2}{n}-\frac{2}{2n-1}\right](\sin\vartheta_1)^{2n-3}d\vartheta_1 = d \left[ \frac{(\sin\vartheta_1)^{2n-2}\cos\vartheta_1}{n(2n-1)}\right] \,, \end{eqnarray} the last integral becomes \begin{eqnarray} && \int_{K\cap\partial B} d \left[ \frac{(\sin\vartheta_1)^{2n-2}\cos\vartheta_1}{n(2n-1)}\cdot \prod_{\ell=2}^{2n-3}(\sin\vartheta_\ell)^{2n-2-\ell}d\vartheta_2\wedge\ldots\wedge d\vartheta_{2n-2}\right]\\ &=& \int_{\partial[K\cap\partial B]} \frac{(\sin\vartheta_1)^{2n-1}\cos\vartheta_1}{n(2n-1)} \prod_{\ell=2}^{2n-3}(\sin\vartheta_\ell)^{2n-2-\ell}d\vartheta_2\wedge\ldots\wedge d\vartheta_{2n-2}\,, \end{eqnarray} where $\partial[K\cap\partial B]$ is a $(2n-3)$-dimensional spherical chain obtained as the intersection of $\partial B$ with the relative boundary of $K$. Now, $\partial[K\cap\partial B]$ is a finite union of hyperplane sections of $\partial B$. Even if one writes down the linear equations of each boundary piece in order to eliminate the variable $\vartheta_1$, the resulting integral is very nasty and I hardly imagine it can be zero!

Could anybody suggest a way out? Thank you in advance.



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