On a sharp Gagliardo–Nirenberg-type inequality

We provide a complete proof of a sharp Gagliardo–Nirenberg-type inequality on the unit interval, in its additive-invariant form \inf_{c\in\mathbb{R}}\|f-c\|_{L^\infty}\le \biggl(\frac{3}{2}\biggr)^{1/3}\bigl(\inf_{c\in\mathbb{R}}\|f-c\|_{L^2}\bigr)^{2/3}\|f'\|_{L^\infty}^{1/3}, which is motivated by the quantitative stability analysis in optimal transport theory. The proof employs the one-dimensional area formula together with rather elementary calculations.