Equivalence of the Apollonian and Its Inner Metric

Table 1

Inequalities between the metrics 𝛼_𝐷, 𝑗_𝐷, 𝛼_𝐷, and 𝑘_𝐷. The subscripts are omitted for clarity with the understanding that every metric is defined in the same domain. The A-column refers to whether the inequality can occur in simply connected planar domains, the B-column refers to whether it can occur in proper subdomains of ℝ^𝑛.

No.	Inequality	A	B	No.	Inequality	A	B
(1)	𝛼≈𝑗≈𝛼≈𝑘	+	+	(7)	𝛼≈𝑗≪𝛼≪𝑘	−	−
(2)	𝛼≪𝑗≈𝛼≈𝑘	−	−	(8)	𝛼≪𝑗≪𝛼≪𝑘	−	−
(3)	𝛼≈𝑗≈𝛼≪𝑘	−	−	(9)	𝛼≈𝛼≪𝑗≈𝑘	−	+
(4)	𝛼≪𝑗≈𝛼≪𝑘	−	−	(10)	𝛼≪𝛼≪𝑗≈𝑘	−	+
(5)	𝛼≈𝑗≪𝛼≈𝑘	+	+	(11)	𝛼≈𝛼≪𝑗≪𝑘	−	−
(6)	𝛼≪𝑗≪𝛼≈𝑘	+	+	(12)	𝛼≪𝛼≪𝑗≪𝑘	−	−