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

The sharpness of the integral inequality $\int_\gamma\sqrt{k_1^2+k_2^2+k_3^2} ds>2\pi$ for closed curves with nowhere 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 $\int_\gamma\sqrt{k_1^2+k_2^2+k_3^2} ds\geq 2\sqrt{5}\pi$.