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2025
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| Online-Zugang: | https://doi.org/10.5281/zenodo.17032125 |
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| _version_ | 1866901513046065152 |
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| author | Sakib, S M Nazmuz |
| author_facet | Sakib, S M Nazmuz |
| contents | <p>We present a simple, foundational, and sharp extremal principle in the line geometry of a<br>reference triangle ABC. Writing any line L in trilinear form u x + v y + w z = 0 (with x, y, z<br>the directed distances to the sidelines), we define for p > 1 the functional<br>Jp(L) = ha |u|p + hb |v|p + hc |w|p, under the normalization |u| + |v| + |w| = 1,<br>where ha, hb, hc are the altitudes to the corresponding sides. The S M Nazmuz Sakib Lp<br>Line Extremal Principle shows:<br>Jp(L) ≥<br><br>h− 1<br>p−1<br>a + h− 1<br>p−1<br>b + h− 1<br>p−1<br>c<br>1−p<br>,<br>with equality for a unique line Lp having trilinear coefficients u : v : w = h− 1<br>p−1<br>a : h− 1<br>p−1<br>b :<br>h− 1<br>p−1<br>c . The result yields a one-parameter continuum of distinguished lines, closed-form best<br>constants, and clean special cases (e.g. at p = 2, the sharp constant equals the inradius r).<br>The proof uses only trilinear coordinates and strict convexity/H¨older-type minimization. We<br>also include illustrative figures, corollaries, and a numerical example.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17032125 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | S M Nazmuz Sakib Lp Line Extremal Principle in Triangle Line Geometry: A Sharp One-Parameter Family with Closed-Form Constants and a Unique Minimizing Line Sakib, S M Nazmuz <p>We present a simple, foundational, and sharp extremal principle in the line geometry of a<br>reference triangle ABC. Writing any line L in trilinear form u x + v y + w z = 0 (with x, y, z<br>the directed distances to the sidelines), we define for p > 1 the functional<br>Jp(L) = ha |u|p + hb |v|p + hc |w|p, under the normalization |u| + |v| + |w| = 1,<br>where ha, hb, hc are the altitudes to the corresponding sides. The S M Nazmuz Sakib Lp<br>Line Extremal Principle shows:<br>Jp(L) ≥<br><br>h− 1<br>p−1<br>a + h− 1<br>p−1<br>b + h− 1<br>p−1<br>c<br>1−p<br>,<br>with equality for a unique line Lp having trilinear coefficients u : v : w = h− 1<br>p−1<br>a : h− 1<br>p−1<br>b :<br>h− 1<br>p−1<br>c . The result yields a one-parameter continuum of distinguished lines, closed-form best<br>constants, and clean special cases (e.g. at p = 2, the sharp constant equals the inradius r).<br>The proof uses only trilinear coordinates and strict convexity/H¨older-type minimization. We<br>also include illustrative figures, corollaries, and a numerical example.</p> |
| title | S M Nazmuz Sakib Lp Line Extremal Principle in Triangle Line Geometry: A Sharp One-Parameter Family with Closed-Form Constants and a Unique Minimizing Line |
| url | https://doi.org/10.5281/zenodo.17032125 |