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A nearest-point map is canonical only when existence and uniqueness are earned. Closed convex targets in CAT(0) spaces provide that control; local convexity alone does not automatically supply it.
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Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Define the projection
Route 2 · Work the metric ledger
Route 3 · Build the retraction
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE PROJECTION
1. Start with the theorem that makes projection canonical
Let X be a complete CAT(0) space and C a nonempty closed convex subset. Every point x in X has a unique nearest point π_C(x) in C. The map π_C is the metric projection. Closedness supports existence, strict convexity of squared distance supports uniqueness, and global convexity keeps the geodesic between target points inside C.
Remove one hypothesis and the conclusion can fail. An open target can have an unattained distance infimum. A nonconvex target can have two equally near points. A projection chosen by a tie-breaker may jump.
CHAPTER 2 OF 12 · DEFINE THE PROJECTION
2. Separate local convexity from global convexity
A subset is locally convex when every point has a neighbourhood in which nearby target points join by ambient geodesics that stay in the subset. This is local information. A global nearest-point theorem needs a target that is globally convex or an additional theorem whose hypotheses promote local convexity to the required global statement.
Connectedness alone is not that theorem. Nor is a locally straight drawing. State completeness, connectedness, local geometry and any covering or simple-connectivity condition used in the local-to-global step.
CHAPTER 3 OF 12 · WORK THE METRIC LEDGER
3. Work a projection ledger on a tree
Let X be a metric tree shaped like the letter Y, with centre o and three unit branches ending at a, b and c. Take C to be the geodesic segment from a to b. It is closed and convex. A point x halfway along the third branch has nearest point o, so π_C(x)=o.
Move x toward c; its projection remains o. Move a point along the a-branch; it already lies in C and is fixed. For any two points, projecting to C never increases their distance.
| Point or pair | Before projection | After projection | Reading |
|---|---|---|---|
| x on c-branch | outside C | o | unique nearest point |
| y on a-branch | inside C | y | target fixed |
| x and y | path passes through o | o and y | distance does not increase |
CHAPTER 4 OF 12 · WORK THE METRIC LEDGER
4. Use the one-Lipschitz property
For closed convex C in a CAT(0) space, d(π_C(x),π_C(y)) ≤ d(x,y). Small movement in the ambient space cannot be amplified by projection. This continuity is stronger than merely saying a nearest point exists.
The inequality supports stable retractions and algorithms, but it does not say distance to C is preserved. Many ambient points can collapse to one gate, as the entire third branch does in the Y-tree.
CHAPTER 5 OF 12 · WORK THE METRIC LEDGER
5. Check equivariance from uniqueness
Suppose a group G acts on X by isometries and preserves C. For g in G, the point gπ_C(x) lies in C and is nearest to gx because distances are preserved. Uniqueness then gives π_C(gx)=gπ_C(x). The projection is G-equivariant.
Both conditions matter. If C is not invariant, gπ_C(x) may leave C. If the action is not isometric, nearest points need not be carried to nearest points. Equivariance is derived from geometry, not added as a label.
CHAPTER 6 OF 12 · BUILD THE RETRACTION
6. Build a strong deformation retraction
For each x, join x to π_C(x) by the unique CAT(0) geodesic and let H(x,t) be the point at fraction t. Then H(x,0)=x and H(x,1)=π_C(x). If x lies in C, π_C(x)=x, so H fixes C for every t.
Continuous dependence of CAT(0) geodesics and the one-Lipschitz projection make H continuous. If the action is isometric and C invariant, uniqueness of geodesics also gives H(gx,t)=gH(x,t). Thus the construction is an equivariant strong deformation retraction.
CHAPTER 7 OF 12 · BUILD THE RETRACTION
7. Changed context: a locally convex loop
In the Euclidean plane, the unit circle is locally arc-like and may look convex at a tiny drawing scale, but it is not geodesically convex: the straight segment between opposite points leaves the circle. A point at the centre has every point of the circle at the same distance, so nearest-point projection is not unique.
The radial formula x/||x|| defines a retraction only away from the centre and cannot be extended there continuously. Local appearance did not supply a global canonical projection.
CHAPTER 8 OF 12 · BUILD THE RETRACTION
8. Apply the method to cores with care
Convex fibres in a product of trees are important, but fibrewise convexity does not automatically prove that the entire chosen subset is globally convex in the product metric. To obtain a nearest-point retraction onto an augmented core, cite the actual CAT(0), closedness, convexity or explicit flow theorem used for that object.
Continue through Augmented-Core Flow, CAT(0) Links and Contractibility, Core Slices, Convex Fibres and One-Dimensional Compatibility and Guirardel Cores, Heavy Quadrants and Intersection Numbers before replacing a proved deformation with an assumed projection.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
A retract need not be a deformation retract, and a deformation retract need not arise from nearest-point projection. Contractible subsets need not be convex. CAT(0) is a metric property; changing the metric can change geodesics and projections while preserving the underlying topology.
Projection onto a nonconvex closed set can be multivalued. Local convexity needs a precise ambient geodesic meaning. An equivariant map does not imply the group action has a fixed point, and a strong deformation retraction does not preserve all metric distances.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Build an evidence check
State X, its metric, completeness and CAT(0) justification. State C, nonemptiness, closedness, global convexity or the exact local-to-global theorem invoked. Prove existence and uniqueness of π_C and identify the norm or metric behind “nearest.”
For equivariance, record the group action, verify isometries and C-invariance, then check π_C(gx)=gπ_C(x). For the homotopy, verify endpoints, fixed target, continuity and equivariance. Use a counterexample whenever one hypothesis is removed.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Questions readers should ask
Does every closed subset of a CAT(0) space have a unique nearest point?
No. Convexity is essential; a closed nonconvex set can have several equally near points.
Is nearest-point projection always distance preserving?
No. It is nonexpansive in this setting and can collapse whole regions.
Why does equivariance follow?
An isometry preserving the target carries a nearest point to a nearest point, and uniqueness forces the two candidate projections to agree.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue through verified routes
Begin with the Topology root, Gold Standard of Education and How X Works. Continue through Compatibility Fans, Irreducible Trees and Length-Function Sums, Compatibility JSJ Trees, Common Refinements and Universal Compatibility and Deformation Spaces of Trees, Collapse Moves and Non-Unique Splittings.
Primary routes include Guirardel’s Core and intersection number for group actions on trees, Guirardel and Levitt’s Deformation spaces of trees, and Bridson and Haefliger’s Metric Spaces of Non-Positive Curvature. The Y-tree and circle transfer checks are original teaching examples.
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