- CARMA COLLOQUIUM
- Speaker: Prof Rafa Espinola, Department of Mathematical Analysis, University of Seville
- Title: Ptolemy vs. CAT(0) spaces
- Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle
- Time and Date: 4:00 pm, Thu, 15th Sep 2011
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). If the quadrilateral is given with its four vertices $A$, $B$, $C$, and $D$ in order, then the theorem states that:
$$|AC| \cdot |BD| = |AB| \cdot |CD| + |AD| \cdot |BC|.$$
Furthermore, it is well known that in every Euclidean (or Hilbert) space $H$ we have that
$$||x - y|| \cdot ||z - w|| \leq ||x - z|| \cdot ||y - w|| + ||z - y|| \cdot ||x - w||$$
for any four points $w, x, y, z \in H$. This is the classical Ptolemy inequality and it is well-known that it characterizes the inner product spaces among all normed spaces. A Ptolemy metric space is any metric space for which the same inequality holds, replacing norms by distances, for any four points.
CAT(0) spaces are geodesic spaces of global nonpositive curvature in the sense of Gromov. Hilbert spaces are CAT(0) spaces and, even more, CAT(0) spaces have many common properties with Hilbert spaces. In particular, although a Ptolemy geodesic metric space need not be CAT(0), any CAT(0) space is a Ptolemy metric space. In this expository talk we will show some recent progress about the connection between Ptolemy metric spaces and CAT(0) spaces.
- [Permanent link]