Croke isoperimetric inequality
From Encyclopedia of Mathematics
Let be a bounded domain in a complete Riemannian manifold
with smooth boundary
. A unit vector
is said to be a direction of visibility at
if the arc of the geodesic ray
from
up to the first boundary point
is the shortest connection between the points
and
, i.e.
. Let
be the set of directions of visibility at
and define the minimum visibility angle of
by
![]() |
where .
Then the following inequalities hold:
![]() | (a1) |
![]() | (a2) |
Both inequalities (a1) and (a2) are sharp in the sense that equality holds if and only if and
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 |
How to Cite This Entry:
Croke isoperimetric inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Croke_isoperimetric_inequality&oldid=17819
Croke isoperimetric inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Croke_isoperimetric_inequality&oldid=17819
This article was adapted from an original article by H. Kaul (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article