{
  "abstract": "The distance between two points is provided by the length of their test connecting path. However, there is no unique way to attain this path in the space $R^2$, which is split by a straight line $L$ into two regions, $Ω_p$ and $Ω_q$, with $\\ell_{p}$- and $\\ell_q$-norms, respectively, where $1<p\\leq q $ and $L\\subsetΩ_q$. In this paper, given a point $P\\in Ω_p$, the locus of points $Q\\inΩ_p$, called $p$-curve, is found, such that there are two ways to attain the distance between $P$ and $Q$. For each $P\\in Ω_p$, there are two $p$-curves which are two branches of two different $p$-parabola. These two branches split $Ω_p$ into three subregions using a three-link shortest path in two of these regions. The implicit equation of each $p$-parabola is given. Furthermore, by using the $p$-curve, an improved algorithm, called P-MFP, based on the MFP algorithm, is developed in order to solve the single-facility location problem in two regions with $\\ell_{p}$ and $\\ell_{q}$-norms. The results obtained with the new algorithm are better than those obtained with the MFP algorithm.ike subheadings, citations, or equations are permitted.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01340",
  "author_total": 4,
  "authors": [
    {
      "name": "Luis Franco"
    },
    {
      "name": "Francisco Velasco"
    },
    {
      "name": "Francisco J. Ortega-Irizo"
    },
    {
      "name": "Luis Gonzalez-Abril"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/esktmk3f6ge7r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01340",
  "fulltext_url": null,
  "id": "esktmk3f6ge7r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01340",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1407|Facility Location and Emergency Management"
    ]
  ],
  "openalex": "W7216038506",
  "pdf_url": "https://arxiv.org/pdf/2610.01340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/esktmk3f6ge7r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01340"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "P-curve: An improved solution for the single-facility location problem in two regions with $\\ell_{p}$- and $\\ell_{q}$-norms",
  "title_text": "P-curve: An improved solution for the single-facility location problem in two regions with ℓ_p- and ℓ_q-norms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/esktmk3f6ge7r",
  "venue": "arXiv",
  "versions": []
}