TITLE: On closed simple curve with curvature at most 1
QUESTION [17 upvotes]: I am looking for the reference to the following theorem.
I have to apply a similar statement, and it would be nice to trace the source.
Please note, I know few proofs in fact it is Problem 3 in my collection of exercises. 

Theorem. Let $\gamma$ be a closed simple plane curve with curvature at most 1. Then the  disc bounded by $\gamma$ contains a circle of radius $1$.

Here is an illustration for those who are lazy to read.

REPLY [7 votes]: Here is summary of letter from Yurii Nikonorov, it gives complete answer.
The statement is called Pestov--Ionin theorem:

Пестов Г., Ионин В. О наибольшем круге, вложенном в замкнутую кривую // Докл. АН СССР. – 1959. – 127, № 6. – С. 1170–1172.

An alternative proof via curve shortening flow was given by Pankrashkin:

Pankrashkin, Konstantin An inequality for the maximum curvature through a geometric flow. Arch. Math. (Basel) 105 (2015), no. 3, 297–300.