Pith Number

pith:F3SH3DIF

pith:2024:F3SH3DIFZSKFKMMNOP5XXAHQT4

not attested not anchored not stored refs pending

Tropical Fermat-Weber Polytropes

David Barnhill, John Sabol, Keiji Miura, Ruriko Yoshida

The set of all Fermat-Weber points under the tropical metric forms a polytrope for any finite sample.

arxiv:2402.14287 v6 · 2024-02-22 · math.CO

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{F3SH3DIFZSKFKMMNOP5XXAHQT4}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

the set of all possible Fermat--Weber points forms a polytrope. This follows from the fact that our location problem turns out to be dual to a particular minimum-cost flow problem.

C2weakest assumption

The location problem is dual to a particular minimum-cost flow problem whose optimal solutions exactly parametrize the Fermat-Weber set (abstract, paragraph 3). If this duality does not hold for the chosen tropical dissimilarity measure, the polytrope claim collapses.

C3one line summary

The set of tropical Fermat-Weber points forms a polytrope via duality to min-cost flow, with a gradient-descent algorithm to locate it.

Receipt and verification

First computed	2026-06-26T01:15:41.984276Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

2ee47d8d05cc9455318d73fb7b80f09f08b1b435257b937eba3662baa0dd03aa

Aliases

arxiv: 2402.14287 · arxiv_version: 2402.14287v6 · doi: 10.48550/arxiv.2402.14287 · pith_short_12: F3SH3DIFZSKF · pith_short_16: F3SH3DIFZSKFKMMN · pith_short_8: F3SH3DIF

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/F3SH3DIFZSKFKMMNOP5XXAHQT4 \
  | 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: 2ee47d8d05cc9455318d73fb7b80f09f08b1b435257b937eba3662baa0dd03aa

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "ea3603f9133c24e3570c1dbb0a86da73edd5b7f1cdfa3c5a37f8b1af33a9d979",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2024-02-22T05:00:00Z",
    "title_canon_sha256": "ea8c9384e885d89bad97e57c7c83e337d069327f192eb217edf19e269f671292"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2402.14287",
    "kind": "arxiv",
    "version": 6
  }
}