A logarithmic mean and intersections of osculating hyperplanes
Classical Analysis and ODEs
2014-07-30 v2
Abstract
We discuss a special case of a family of means defined using intersections of osculating hyperplanes to curves in R^n. Let C be the curve in R^n with vector equation x_{k}(t)=t(ln t)^{k-1},k=1,...,n. Let O_{k} be the osculating hyperplane to C at a_{k},k=1,...,n. Then we show that O_{1},...,O_{n} have a unique point of intersection, P=(i_{1},...,i_{n}), and in particular, i_{1} equals the logarithmic mean in n variables of Neuman.
Keywords
Cite
@article{arxiv.1404.1489,
title = {A logarithmic mean and intersections of osculating hyperplanes},
author = {Alan Horwitz},
journal= {arXiv preprint arXiv:1404.1489},
year = {2014}
}
Comments
Minor revisions and a better worded abstract. 19 pages