91Ó°ÊÓ

The triangle inequality is a crucial property that any valid metric must satisfy. Essentially, it states that for any three points in a metric space, the direct path between two points is never longer than making a detour through a third point. In mathematical terms, for three points \( x, y, z \), this is expressed as:\[d(x, z) \leq d(x, y) + d(y, z).\]This condition ensures that the metric behaves in a consistent and predictable manner, mimicking the properties of distances in Euclidean space. In the context of our exercise, the function \(d(x, y)\) defined for the disk only needs to satisfy this condition under relevant circumstances, such as points lying on the same line through the origin or otherwise.

• If all three points lie on the same line through the origin, the standard Euclidean triangle inequality \( \|x-z\| \leq \|x-y\| + \|y-z\| \) naturally applies.
• If not, the metric is structured such that the sum of segments' lengths will inherently satisfy \( d(x, z) \leq d(x, y) + d(y, z) \), owing to being the summation of Euclidean norms.

Observations like these showcase the importance of the triangle inequality in confirming \(d\) as a valid metric.

A discrete metric is a simple yet interesting form of a metric where distance between any two distinct points is always the same, specifically set to 1. This means:

• \( d(x, y) = 0 \) if \( x = y \)
• \( d(x, y) = 1 \) if \( x eq y \)

In our exercise, we are asked to show that the given metric, when restricted to the circle \( S^1 \), acts like a discrete metric. Since all points on \( S^1 \) have a norm of 1, \( d(x, y) = \|x\| + \|y\| = 2 \) for any distinct points, unless they lie directly on a line through the origin where \( \|x-y\| \) would apply. However, since any two points on \( S^1 \) can't directly lie on a line through the origin (as it would imply they are identical or antipodal in infinite space), the distance is simplified as 1 for distinct points on \( S^1 \), thus mimicking the discrete metric on this subset. This unique characteristic aids in clearly identifying distinct points on the circle, simplifying many analytical processes.

To induce a metric on a subspace means creating a new metric based on an existing one, but specifically applied to a smaller set within the larger space. The induced metric essentially restricts the rules of the original metric to the subspace while potentially taking on new characteristics unique to that subspace.

In the given exercise, \(d\) is tested for being an induced metric on the subspace \( S^1 \), which is the circle where each point \( x \) has a norm of 1. When applying the general rule of \(d\) onto \( S^1 \), we find that:

• For \( x, y \in S^1 \), if \( x eq y \), the metric naturally transitions to the discrete form, given \( d(x, y) = 2 \).
• If \( x = y \), then of course \( d(x, y) = 0 \).

Through this restriction, \(d\) forms a discrete metric on \( S^1 \), highlighting the versatility of how metrics can be tailored and adapted to new constraints or conditions established by the subspace. Understanding this concept helps in seeing how larger spaces interact with and influence characteristics of their subspaces efficiently.