On the optimality of one integral inequality for closed curves in $\mathbb R^4$

Abstract

The optimality of the integral inequality $\int\limits_\gamma\sqrt{k_1^2+k_2^2+k_3^2}ds>2\pi$ for closed curves with non-vanishing curvatures in $\mathbb R^4$ is discussed. We prove that an arbitrary closed curve of constant positive curvatures in $\mathbb R^4$ satisfies the inequality $\int\limits_\gamma\sqrt{k_1^2+k_2^2+k_3^2}ds \geq 2\sqrt{5}\pi$.