The modified horizontal derivative and some of its applications (Q1921811)

The author studies the following theorem of Mukhometov. A simple metric \(g\) on a compact manifold with boundary is uniquely determined in its conformal class by distances among boundary points. Here \(g\) is called `simple' if any two points are joined by a unique geodesic. The author obtains a new estimate of stability and a kinematic formula for the volume of \(g\) which is linear in distances among boundary points. The new notion of modified horizontal derivative allows the author to obtain a variety of Mukhometov-type identities.