Thanks for your example! I have been thinking about it. Your observation is correct and the surface of the sphere is a metric. The ratio of radius to circumfer

Thanks for your example! I have been thinking about it. Your observation is correct and the surface of the sphere is a metric. The ratio of radius to circumference is not constant with that metric though so I feel like something should disqualify it. But I am not sure how. So I think your observation shows that we need a stronger constraint than just being a metric. Other commenters have hinted that you need a normed vector space but I am not sure if that's sufficient.