The Riesz kernels do not give rise to higher dimensional analogues of the Menger-Melnikov curvature
The Menger-Melnikov curvature \(c(z_1,z_2,z_3)\) can be evaluated from the identity, \[ c(z_1,z_2,z_3)^2=\sum_\sigma {1\over (z_{\sigma(1)}-z_{\sigma(3)}) \overline{(z_{\sigma(2)}-z_{\sigma(3)})}} \] where the sum is taken over all permutations of \(\{1,2,3\}\). This curvature has proved useful in connections with the Cauchy kernel and analytic capacity problem. The author investigates the possibility to obtain a similar formula for higher dimensional cases. The main result is that it does not exist a generalization of the following type, where \(H\) is the group of permutations of \(\{0,1,\dots k\}\), \(x^{(i)}\) is the \(i\)-th coordinate of \(x\in{\mathbb R}^n\), and \(a\) is a real valued function on \(\{1,\dots ,n\}^k\): \[ P_{m,k,n}(x_0,x_1,\dots,x_k)=\sum_{\gamma\in H} \sum_{j_1 =1}^n \cdots \sum_{j_k=1}^n {a(j_1,\dots ,j_k)\Pi_{i=1}^k \big(x_{\gamma(i)}^{(j_i)}-x_{\gamma(0)}^{(j_i)}\big) \over \Pi_{i=1}^k|x_{\gamma(i)}-x_{\gamma(0)}|^{m+1}}. \] The Menger-Melnikov curvature is of this form. The main result: Let \(n>2\), if \(m>2\), or \(k\not= 2\) then \(P_{m,k,n}\) is not a curvature function of the Melnikov type (meaning that \(P_{m,k,n}\) is not nonnegative or identically zero).
scientific article; zbMATH DE number 1331815