Shape (In)dependent Inequalities for Triangleland’s Jacobi and Democratic-Linear Ellipticity Quantitities

Abstract Sides and medians are both Jacobi coordinate magnitudes. These furthermore enter equably into the spherical coordinates on Kendall’s shape sphere and the Hopf coordinates. This motivates treating medians on the same footing as sides in the geometry of triangles and the ensuing Shape Theory. In this article, we consequently reformulate inequalities for the medians in terms of shape quantities, and proceed to find inequalities bounding the mass-weighted Jacobi coordinates. This work moreover identifies the 4/3 – powers of which occur frequently in the theory of medians – as the ratio of Jacobi masses.

One of the Hopf coordinates is tetra-area. Another is anisoscelesness, which parametrizes whether triangles are left-or-right leaning as bounded by isoscelesness itself. The third is ellipticity, which parametrizes tallness-or-flatness of triangles as bounded by regular triangles. Whereas tetra-area is clearly cluster-choice invariant, Jacobi coordinates, anisoscelesness and ellipticity are cluster-choice dependent. And yet they can be ‘democratized’ by averaging over all clusters. Democratized ellipticity moreover trivializes, due to ellipticity being the difference of base-side and median second moments, whose averages are equal to each others. Thus we introduce a distinct linear ellipticity’ quantifier of tallness-or-flatness of triangles whose democratization is nontrivial, and find inequalities bounding this.

Some of this article’s inequalities are shape-independent bounds. Others’ bounds depend on the isoperimetric ratio and arithmetic-to-geometric mean ratio of the sides: shape variables.