Optimal Finsler-Hadwiger inequalities

TL;DR

This paper derives the sharpest inequalities relating a triangle's area, perimeter squared, and isoperimetric deficit using geometric optimization techniques.

Contribution

It explicitly finds the optimal Finsler-Hadwiger inequalities, improving understanding of geometric bounds between key triangle measures.

Findings

Explicit sharp inequalities between area, perimeter squared, and isoperimetric deficit.

Use of Blaschke-Santaló diagrams and optimization methods to establish bounds.

Enhanced geometric inequalities for triangles.

Abstract

Various inequalities exist between the area of a triangle, the perimeter squared $(a+b+c)^2$ and the isoperimetric deficit $Q=(a-b)^2+(b-c)^2+(c-a)^2$. The direct and reverse Finsler-Hadwiger inequalities correspond to the best linear inequalities between the three quantities mentioned above. In this paper, the sharpest inequalities between these three quantities are found explicitly. The techniques used involve Blaschke-Santal\'o diagrams and constrained optimization problems.