Pith Number

pith:H3GXZRPL

pith:2026:H3GXZRPL7QSV3OU6WKYGPROIHB

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Retractions by Alternating Projections

Shixiang Chen, Wen Huang, Yixiao He

Alternating projections between two cleanly intersecting manifolds induce a retraction on their intersection.

arxiv:2605.17384 v1 · 2026-05-17 · math.OC

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Record completeness

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Claims

C1strongest claim

under the assumption that two C^{2,1} embedded submanifolds M1, M2 subset R^n intersect cleanly, we show that the associated alternating mapping admits a well-defined local limiting map psi on the intersection manifold M=M1 cap M2, and that psi is a retraction on M. If, in addition, M1 and M2 are C^{3,1}, then psi is a second-order retraction.

C2weakest assumption

The two manifolds are C^{2,1} embedded submanifolds of R^n that intersect cleanly (as stated in the abstract as the key assumption enabling the local limiting map and retraction property).

C3one line summary

For two C^{2,1} embedded submanifolds intersecting cleanly, the alternating projection mapping induces a retraction on the intersection, which is second-order if the manifolds are C^{3,1}.

References

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[1] ¨Uber einige abbildungsaufgaben.Gesammelte Mathematische Abhandlungen 11, pages 65–83, 1869

[2] John von Neumann.Functional Operators. II. The Geometry of Orthogonal Spaces. Annals of Mathematics Studies, 22. Princeton University Press, Princeton, NJ, 1950 1950

[3] The method of successive projections for finding a common point of convex sets 1965

[4] The method of projections for finding the common point of convex sets.USSR Computational Mathematics and Mathematical Physics, 7(6):1–24, 1967 1967

[5] On the convergence of von neumann’s alternating projection algorithm for two sets.Set-Valued Analysis, 1(2):185–212, 1993 1993

Formal links

2 machine-checked theorem links

Cited by

1 paper in Pith

Optimization over the intersection of manifolds

Receipt and verification

First computed	2026-05-20T00:03:55.875436Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3ecd7cc5ebfc255dba9eb2b067c5c838523952022ab73d85eba0807b0027d247

Aliases

arxiv: 2605.17384 · arxiv_version: 2605.17384v1 · doi: 10.48550/arxiv.2605.17384 · pith_short_12: H3GXZRPL7QSV · pith_short_16: H3GXZRPL7QSV3OU6 · pith_short_8: H3GXZRPL

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/H3GXZRPL7QSV3OU6WKYGPROIHB \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 3ecd7cc5ebfc255dba9eb2b067c5c838523952022ab73d85eba0807b0027d247

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "02354881a377d03a7d4983eeeea4442761671fe6848c08361f82533ed5b38ed7",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-17T11:03:31Z",
    "title_canon_sha256": "10887c2dcce6d2ec16083af533adf8e420f65843fa912282c03c7b3aa9939e76"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17384",
    "kind": "arxiv",
    "version": 1
  }
}