On the Grace-Danielsson inequality for tetrahedra

TL;DR

The paper provides a new proof of the Grace-Danielsson inequality for tetrahedra by expressing the difference of squared sides as a sum of nonnegative terms, and characterizes degenerate cases like infinite triangular prisms.

Contribution

It introduces a novel proof of the inequality and characterizes degenerate tetrahedra, including conditions for equality and examples.

Findings

Difference expressed as sum of nonnegative terms

Characterization of infinite triangular prisms as degenerate tetrahedra

Conditions for equality in the inequality

Abstract

The difference between the (squared) sides of the Grace-Danielsson inequality for tetrahedra will be represented as a sum of two nonnegative terms. This gives another proof of the inequality. Examining the denominator allows us to characterize the infinite triangular prism as a degenerate tetrahedron. We give conditions for equality (for a zero gap) as well, and some examples are included.