No abstract

No abstract

A metric space (X, d) is called ptolemaic or short a PT space, if for all quadruples of points x, y, z, w ∈ X the Ptolemy inequality |xy| |zw| ≤ |xz| |yw| + |xw| |yz|(1) holds, where |xy| denotes the distance d(x, y).We prove a flat strip theorem for geodesic ptolemaic spaces. Two unit speed geodesic lines c 0 , c 1 : 1], such that the boundary curves are parallel geodesic lines, then X is isometric to a flat strip R × [0, a] ⊂ R 2 with its euclidean metric.We became interested in ptolemaic metric spaces because of their relation to the geometry of the boundary at infinity of CAT(−1) spaces (compare [FS1], [BS]). We therefore think that these spaces have the right to be investigated carefully.Our paper is a contribution to the following question Q: Are proper geodesic ptolemaic spaces CAT(0)-spaces?We give a short discussion of this question at the end of the paper in section 5. Main ingredients of our proof is a theorem of Hitzelberger and Lytchak [HL] about isometric embeddings of geodesic spaces into Banach spaces and the Theorem of Schoenberg [Sch] characterizing inner product spaces by the PT inequality. PreliminariesIn this section we collect the most important basic facts about geodesic PT spaces which we will need in our arguments. If we do not provide proofs in this section, these can be found in [FLS], [FS2].Let X be a metric space. By |xy| we denote the distance between points x, y ∈ X. We will always parametrize geodesics proportionally to arclength.

The recent financial crisis revealed that banks, especially these large and complex banks, are opaque to be monitored by regulators. In an ideal world, regulators are hoping all banks to be self-disciplined. That will reduce a lot burdens for regulators. However, in practice, it is not always the case as there will be by nature information asymmetric or information frictions between banks and regulators. We develop a tractable model to study how banks respond to capital requirements that are based on a self-assessment result about banks' riskiness, and derive the policy and wealth implications. We use the model to characterize the optimal requirements, and to study the trade-offs a regulator faces in making efforts to ensure bank's self-assessment more accurate or in disclosing the inspection results to public.

No abstract