A subset of a metric space is called a Чебышёв (Chebyshev) set if for every there exists such that for all . In words, for each point of the space there is a well-defined nearest point of . We can then define the nearest point projection by sending to . This is consistent with the notion of orthogonal projection onto a subspace of a Euclidean space. More generally, every closed convex subset of a Hilbert space is a Чебышёв set. Whether the converse is true is **unknown**. It’s been proved in finite dimensions and in some other special cases, such as for weakly closed sets. (Yes, being **weakly **closed is a **stronger **property than being closed…)

There isn’t much one can prove about Чебышёв sets in general metric spaces. They must be closed, and that’s about it. Let’s specialize to being a real Banach space, and a closed subspace thereof. There are two ways in which could fail to be Чебышёв:

- the infimum is attained for more than one point .
- the infimum is not attained at all.

Examples of the first kind are easy to produce. Let be the plane with the taxicab norm . The point is at distance from the line , and this distance is realized by any point with .

It’s also easy to see exactly how we can avoid such examples: assume that the space is *strictly convex*, i.e., its unit sphere does not contain any line segments.

Examples of the second kind are not as simple. To attain the infimum, it suffices to find a cluster point of a minimizing sequence . So if the space is reflexive, we can use the weak compactness of closed balls [note in passing: **weak **compactness **is weaker** than compactness] to obtain a weakly convergent minimizing sequence. Its limit also belongs to and minimizes the distance.

The converse is also true: if every closed subspace is Чебышёв, then is reflexive. Indeed, by the James theorem every non-reflexive space has a linear functional that is not norm-attaining, that is, for all . Let . I claim that has no nearest points to any point .

Fix a point and suppose that is the corresponding nearest point. We may assume . Since , there exists a unit vector such that . The vector belongs to and , a contradiction.

For a concrete example, take and . One can check that the unit vector has , and this distance is **not attained**. The sequence has , but contains no weakly convergent subsequence. Its weak* limit is , which is outside of . Like this:

Just as the name Чебышёв has no nearest point projection onto the Latin alphabet.