I am trying to read the paper "Simple closed geodesics on convex surfaces" by E.Calabi and J. Cao and a certain passage is unclear for me. Before, let me contextualize and set up some notation.

Let $\gamma$ be a simple closed geodesic in a Riemannian sphere $(M^2, g)$ and let $\Omega_j$, $j=1,2$, be the two domains determined by $\gamma$ in $M$. Denote by $UM$ the unit circle bundle of $M$. For fixed $j=1,2$, let $N$ the inwardly pointing unit normal for $\partial \Omega_j$, and let

$$U^+ \partial \Omega_j = \{ v \in T_p M : p \in \partial \Omega_j, \vert v \vert = 1 \text{ and } g(v,N(p)) > 0 \}.$$

For $v \in U^+ \partial \Omega_j$ with $v \in T_p M$, let $\sigma_v$ be a geodesic with $\sigma_v(0) = p$ and $\sigma_v'(0) = v$. Also, let $l(v)$ be the smallest $t > 0$ such that $\sigma_v(t) \in \partial \Omega_j$. Finally, define

$$W(\Omega_j) = \inf \{ l(v) : v \in U^+ \partial \Omega_j\}$$

to be the width of $\Omega_j$.

The authors then write the following:

enter image description here

The problem is that the reference to Santaló's formula in the paper is wrong.

Questions: Could you help me to understand the above inequalities, please? Do you have a reference for the mentioned formula?



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