A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Tserendorj Batbold

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arXiv · 2609.10633

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Tserendorj Batbold

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Abstract

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

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Tserendorj Batbold. 2026-09-09. A Proof of Liu's Conjecture on the Fundamental Triangle Inequality. https://arxiv.org/abs/2609.10633

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