Croke isoperimetric inequality

Let $\Omega$ be a bounded domain in a complete Riemannian manifold $M = M ^ { n }$ with smooth boundary $\partial \Omega$. A unit vector $v \in T _ { p } M$ is said to be a direction of visibility at $p \in \Omega$ if the arc of the geodesic ray $t \mapsto \gamma ( t ) = \operatorname { exp } _ { p } ( t v )$ from $p$ up to the first boundary point $\gamma ( s ) \in \partial \Omega$ is the shortest connection between the points $p$ and $\gamma ( s )$, i.e. $s = \operatorname { dist } ( p , \gamma ( s ) )$. Let $\Omega _ { p } \subset T _ { p } M$ be the set of directions of visibility at $p$ and define the minimum visibility angle of $\Omega$ by

\begin{equation*} \omega = \operatorname { inf } _ { p \in \Omega } \frac { \operatorname { Vol} ( \Omega _ { p } ) } { \alpha ( n - 1 ) }, \end{equation*}

where $\alpha ( k ) = \operatorname{Vol} ( S ^ { k } )$.

Then the following inequalities hold:

\begin{equation} \tag{a1} \frac { \text { Vol } ( \partial \Omega ) } { \text { Vol } ( \Omega ) } \geq \frac { c _ { 1 } } { \operatorname { diam } \Omega } \cdot \omega , \quad c_ { 1 } = \frac { 2 \pi \alpha ( n - 1 ) } { \alpha ( n ) }, \end{equation}

\begin{equation} \tag{a2} \frac { \operatorname {Vol} ( \partial \Omega ) ^ { n } } { \operatorname {Vol} ( \Omega ) ^ { n - 1 } } \geq c _ { 2 } \cdot \omega ^ { n + 1 } , \quad c _ { 2 } = \frac { \alpha ( n - 1 ) ^ { n } } { \left( \frac { \alpha ( n ) } { 2 } \right) ^ { n - 1 } }. \end{equation}

Both inequalities (a1) and (a2) are sharp in the sense that equality holds if and only if $\omega = 1$ and $\Omega$ is a hemi-sphere of a sphere of constant positive curvature.

In the proof of the second inequality, special versions of the Berger inequality and the Kazdan inequality are used.

References

[a1]	I. Chavel, "Riemannian geometry: A modern introduction" , Cambridge Univ. Press (1995)
[a2]	C.B. Croke, "Some isoperimetric inequalities and eigenvalue estimates" Ann. Sci. Ecole Norm. Sup. , 13 (1980) pp. 419–435