On generalized metrics of Vandermonde type

Abstract In a series of papers in the 1960’s, S. Gähler defined and investigated so-called m -metric spaces and their topological properties. An m -metric assigns to any tuple of $$m+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> elements a real value (more generally an element in a partially ordered set) which satisfies the generalized metric axioms of semidefiniteness, symmetry, and simplex inequality. In this contribution we consider a new type of generalized metric which is based on the Vandermonde determinant. We present some remarkable geometric consequences of the corresponding simplex inequality in the complex plane. Then we show that the Vandermonde principle of construction extends to linear spaces of arbitrary dimension by using symmetric multilinear maps of degree $$\frac{1}{2}m(m+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . In particular, we analyze when this generalized metrics has the stronger property of definiteness, i.e. two of its arguments are equal if the metric vanishes. Finally, an application is provided to the m -metric of point sets when driven by the same linear ordinary differential equation.