Contracting the boundary of a Riemannian $2$-disc

Let $D$ be a Riemannian $2$-disc. Denote its area by $A$, its diameter by $d$ and the length
of its boundary by $L$.
We will prove that one can always contract the boundary of $D$ via closed curves (
or even based loops) of length less than $L+200d\max\{1,\ln{\sqrt{A}\over d}\}.$
This answers a twenty-year old question by S. Frankel and M. Katz, a version of which
was asked earlier by M. Gromov.
This result is a joint work with Y. Liokumovich and R. Rotman.
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We will also discuss several related problems and results in metric geometry of Riemannian surfaces.