Various centroids and some characterizations of catenary rotation hypersurfaces

Abstract

We study positive C-1 functions z - f (x), x - (x(1),..., x(n)) defined on the n-dimensional Euclidean space R-n. For x = (x(1),..., x(n)) with nonzero numbers x(1),..., x(n) we consider the rectangular domain I(x) = I(x(1)) x...xI(x(n)) subset of R-n, where /(x(i)) = [0, x(i]) if x(i) > 0 and /(x(i)) = [xi, 0] if x, < 0. We denote by V, S, (xV, ZV), and (xS, xS) the volume of the domain under the graph of z = f (x), the surface area S of the graph of z = f (x), the geometric centroid of the domain under the graph of z = f (x), and the surface centroid of the graph itself over the rectangular domain I (x), respectively. In this paper, first we show that among C-2 functions with isolated singularities, S = kV, k is an element of R characterizes the family of catenary rotation hypersurfaces f (x) = k cosh(r/k), r = vertical bar x vertical bar Next we show that the equality of n coordinates of (xS, xS) and (cV, 2zV) for every rectangular domain I (x) characterizes the family of catenary rotation hypersurfaces among C-2 functions with isolated singularities.