Abstract
AbstractWe prove that given a compact convex non-empty domain $${\mathcal {M}}$$
M
in the plane, a function $$\delta :\partial \mathcal M\times \partial {\mathcal {M}}\rightarrow {\mathbb {R}}_+$$
δ
:
∂
M
×
∂
M
→
R
+
can be extended to a projective metric d on $${\mathcal {M}}$$
M
if and only if $$ \delta (P,R)+\delta (Q,S)-\delta (P,S)-\delta (Q,R)>0 $$
δ
(
P
,
R
)
+
δ
(
Q
,
S
)
-
δ
(
P
,
S
)
-
δ
(
Q
,
R
)
>
0
for any convex quadrangle $$\Box (PQRS)$$
□
(
P
Q
R
S
)
inscribed in $$\partial {\mathcal {M}}$$
∂
M
. Moreover, this extension is unique.
Publisher
Springer Science and Business Media LLC
Reference23 articles.
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2. Alexander, R.: Planes for which the Lines are the shortest paths between points. Ill. J. Math. 22, 177–190 (1978)
3. Beltrami, E.: Risoluzione del problema: riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette, Opere, I (1865), 262–280; Annali di Matematica pura et applicata, serie I, VII (1865), 185–204; https://gallica.bnf.fr/ark:/12148/bpt6k99432q/f287
4. Busemann, H., Kelly, P.J.: Projective Geometry and Projective Metrics. Academic Press, New York (1953)
5. Busemann, H.: The Geometry of Geodesics. Academic Press, New York (1955)
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