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

Abstract

The optimality of the integral inequality $\underset{\gamma }{\int }\sqrt{k_1^2+k_2^2+k_3^2}ds>2\pi$ 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 $\underset{\gamma }{\int }\sqrt{k_1^2+k_2^2+k_3^2}ds\ge 2\sqrt{5}\pi$.