# Vectors and a half-disk of triangle shapes in Ionescu-Weitzenb\"ock's inequality

## TL;DR

This paper presents two new conceptual proofs of Ionescu-Weitzenb"ock's inequality, offering geometric insights and a novel approach using the set of triangles as a half-disk, enriching understanding of this classical inequality.

## Contribution

It introduces two innovative proofs of Ionescu-Weitzenb"ock's inequality, one vector-based with geometric interpretation, and another using properties of triangle sets as a half-disk.

## Key findings

- Vector proof provides geometric interpretation of the inequality
- Second proof uses properties of triangles as a half-disk
- Both proofs offer new perspectives on classical inequality

## Abstract

The aim of this note is to give two new conceptual proofs of Ionescu-Weitzenb\"ock's inequality. The first one, which is a vector proof, provides us a geometric interpretation of the difference between the two sides of this inequality and of two known corollary inequalities in differential geometry. Through a less classical approach, our second proof makes use of the properties of the set of triangles with a fixed "size", seen as a half-disk.