Vol. 101, No. 2, 1982

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Ptolemy’s inequality, chordal metric, multiplicative metric

M. S. Klamkin and A. Meir

Vol. 101 (1982), No. 2, 389–392

Ptolemy’s inequality in R2 states: If A, B, C, D are vertices of a quadrilateral, then

AB ⋅CD  + BC ⋅AD  ≧ AC ⋅BD

with equality only ABCD is a convex cyclic quadrilateral. A real normed linear vector space is called ptolemaic if

∥x − y∥∥z∥+ ∥y− z∥∥x∥ ≧ ∥z − x∥∥y∥

for all x, y and z in the space and it is called symmetric if

∥λx − y∥ = ∥x− λy∥

for all unit vectors x, y and real λ. The equivalence of these two properties of a normed linear space is established and related results concerning distance functions in such spaces are proven.

Mathematical Subject Classification 2000
Primary: 51K05
Secondary: 46B99
Received: 2 May 1980
Published: 1 August 1982
M. S. Klamkin
A. Meir