Minimal properties for convex annuli of plane convex curves. (English) Zbl 0820.52002
Let \(K\) be a fixed closed convex curve in the Euclidean plane with the origin in its interior.
By \(K\)-annulus of centre \(c\) we mean the set of all points between the curves of the form \(\rho K + c\) and \(\sigma K + c\), where \(0 \leq \rho \leq \sigma\). A \(K\)-annulus is said to enclose a convex curve \(C\) if no point of \(C\) is out of the curve \(\sigma K + c\), whereas any point of the curve \(\rho K + c\) is inside \(C\). This \(K\)-annulus is said to bi-enclose \(C\) if \(C\) passes at least four times between the two curves bounding the annulus.
Generalizing results due to T. Bonnesen and I. Vincze on circular annuli, we prove that for any closed convex curve \(C\) there exists a unique \(K\)- annulus bi-enclosing \(C\). Such an annulus minimizes either the difference \(\sigma - \rho\) or the ratio \(\sigma/ \rho\), and it is the only one satisfying both these properties.
Moreover, differently from a Nagy’s result on circular annuli, this \(K\)- annulus does not necessarily have minimal area.
By \(K\)-annulus of centre \(c\) we mean the set of all points between the curves of the form \(\rho K + c\) and \(\sigma K + c\), where \(0 \leq \rho \leq \sigma\). A \(K\)-annulus is said to enclose a convex curve \(C\) if no point of \(C\) is out of the curve \(\sigma K + c\), whereas any point of the curve \(\rho K + c\) is inside \(C\). This \(K\)-annulus is said to bi-enclose \(C\) if \(C\) passes at least four times between the two curves bounding the annulus.
Generalizing results due to T. Bonnesen and I. Vincze on circular annuli, we prove that for any closed convex curve \(C\) there exists a unique \(K\)- annulus bi-enclosing \(C\). Such an annulus minimizes either the difference \(\sigma - \rho\) or the ratio \(\sigma/ \rho\), and it is the only one satisfying both these properties.
Moreover, differently from a Nagy’s result on circular annuli, this \(K\)- annulus does not necessarily have minimal area.
Reviewer: C.Peri (Milano)
MSC:
| 52A10 | Convex sets in \(2\) dimensions (including convex curves) |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
References:
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| [2] | T.Bonnesen und W.Fenchel, Theorie der konvexen Körper. Berlin 1934. English translation: Theory of convex bodies, Moscow-Idaho 1987. · Zbl 0008.07708 |
| [3] | B. Fuglede, Bonnesen’s inequality for the isoperimetric deficiency of closed curves in the plane. Geom. Dedicata38, 283-300 (1991). · Zbl 0723.53001 · doi:10.1007/BF00181191 |
| [4] | J. S. Nagy, Konvexe Kurven und einschließende Kreisringe. Acta Sci. Math. (Szeged)10, 174-184 (1941-1943). |
| [5] | C. Peri, On the minimal convex shell of a convex body. Canad. Math. Bull.36, 466-472 (1993). · Zbl 0817.52007 · doi:10.4153/CMB-1993-062-x |
| [6] | C. Peri, J. M. Wills andA. Zucco, On Blaschke’s extension of Bonnesen’s inequality. Geom. Dedicata48, 349-357 (1993). · Zbl 0787.52004 · doi:10.1007/BF01264078 |
| [7] | C. Peri andA. Zucco, On the minimal convex annulus of a planar convex body. Monatsh. Math.114, 125-113 (1992). · Zbl 0765.52003 · doi:10.1007/BF01535579 |
| [8] | I. Vincze, Über Kreisringe, die eine Eilinie einschließen. Studia. Sci. Math. Hungar.9, 155-159 (1974). · Zbl 0305.52003 |
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