In this paper, we consider a problem in calculus of variations motivated by a quantitative isoperimetric inequality in the plane. More precisely, the aim of this article is the computation of the minimum of the variational problem $$inf_{uinmathcal{W}}frac{displaystyleint_{-pi}^{pi}[(u)^2-u^2]dtheta}{left[int_{-pi}^{pi}|u| dthetaright]^2}$$ where $uin mathcal{W}$ is a $H^1(-pi,pi)$ periodic function, with zero average on $(-pi,pi)$ and orthogonal to sine and cosine.
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