# What integral formula is being used here?

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:

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?

• Jan 25 at 3:54
• @FrancoisZiegler perfect! Forgot to search in Wiki 😅 Jan 25 at 16:39