In this short note we prove that the Winterbottom shape (Winterbottom in Acta Metallurgica 15:303-310, 1967) is a volume-constraint minimizer of the corresponding anisotropic capillary functional.

For a given p-variable mean M:Ip→I (I is a subinterval of R), following (Horwitz in J Math Anal Appl 270(2):499–518, 2002) and (Lawson and Lim in Colloq Math 113(2):191–221, 2008), we can define (under certain assumptions) its (p+1)-variable β-invariant extension as the unique solution K:Ip+1→I of the functional equation K(M(x2,⋯,xp+1),M(x1,x3,⋯,xp+1),⋯,M(x1,⋯,xp))=K(x1,⋯,xp+1),for allx1,⋯,xp+1∈Iin the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions.

In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n-metric which assigns a value to a tuple of n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 2$$\\end{document} points. Some elementary properties of pseudo n-metrics are provided and their construction via exterior products is investigated. We discuss some examples from the geometry of Euclidean vector spaces leading to pseudo n-metrics on the unit sphere, on the Stiefel manifold, and on the Grassmann manifold. Further, we construct a pseudo n-metric on hypergraphs and discuss the problem of generalizing the Hausdorff metric for closed sets to a pseudo n-metric.