Abstract
In this paper, we establish some Bonnesen-style isoperimetric inequalities for plane polygons via an analytic isoperimetric inequality and an isoperimetric inequality in pseudo-perimeters of polygons.
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CHEVEL, I.: Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.
CHEVEL, I.: Riemannian Geometry: A Modern Introduction, Cambridge Press, New York, 1993.
FEJES TÓTH, L.: Lagerungen in der Ebene, auf der Kugel und im Raum, Die Grundlehren der math. Wissenschaften, vol. 65, Springe-Verlag (1972).
CHRIS FISHER, J., RUOFF, D. and SHILLETO, J.: Perpendicular Polygons, Amer. Math. Monthly, Jan. (1985) 23–37.
GROEMER, H.: Stability Properties of Geometric Inequalities, Amer. Math. Monthly, 97(1990)382–394.
GROEMER, H.: Stability Theorems for Convex Domains of Constant Width, Canad. Math. Bull. 31(1988)328–337.
GRUBER, P. M. and WILLS, J. M.: Handbook of Convex Geometry, North-Holland, 1993.
HARDY, G., LITTLEWOOD, J. E. and POLYA, G.: Inequalities, Cambridge Univ. Press, Cambridge, New York, 1951.
KAZARINOFF, M. D.: Geometric Inequalities, New Math. Library, Math. Assoc. of America, 1961.
KU, H. T., KU, M. C. and ZHANG, X. M.: Analytic and Geometric Isopermetric Inequalities, Journal of Geometry, vol.53(1995)100–121.
KU, H. T., KU, M. C. and ZHANG, X. M.: Isoperimetric Inequalities on Surfaces of Constant Curvature, (Preprint).
MACNAB, D. S.: Cyclic Polygons and Related Questions, Math. Gazette, 65(1981)22–28.
MARSDEN, J. E. and TROMBA, A. J.: Vector Calculus, Second Edition, W. H. Freeman and Company, 1981.
MITRINOVIĆ, D. S.: Analytic Inequalities, Springer-Verlag Berlin-Heidelberg-New York, 1970.
OSSERMAN, R.: The Isoperimetric Inequality, Bull. Amer. Math. Soc. 84(1978), 1182–1238.
OSSERMAN, R.: Bonnesen-style Isoperimetric Inequalities, Amer. Math. Monthly, No.1(1979) 1–29.
PERI, W. and ZUCCO: On Blaschke's Extension of Bonnnesen's Inequality, Geometria Dedicata, vol.48, No.3(1993)349.
POLYA, G. and SZEGO, G.: Isoperimetric Inequalities in Mathematical Physics, Ann. of Math. Studies, No.27, Princeton Univ., Princeton, 1951.
ROBBINS, DAVID P.: Area of Polygons Inscribed in a Circle, Amer. Math. Monthly, [June–July] (1995)523–530.
SANGWINE-YAGER, J. R.: Bonnesen-style Inequalities for Minkowski Relative Geometry, Trans. AMS., vol.307(1988)373–382.
TANG, D.: Discrete Wirtinger and Isoperimetric Type Inequalities, Bull. Austral. Math. Soc. 43(1991),467–474.
TEISSIER, B.: Bonnesen-type Inequalities in Algebraic Geometry, Seminar on Differential Geometry, Edited by S. T. Yau, Princeton Univ. Press and Univ. of ToKyo press, Princeton, 1982.
ZHANG, X. M.: A Refinement of the Discrete Wirtinger Inequality, Journal of Mathematical Analysis and Applications, to appear.
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1991 Mathematics Subject Classification 51M10, 51M25,52A40,26D10.
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Zhang, XM. Bonnesen-style inequalities and pseudo-perimeters for polygons. J Geom 60, 188–201 (1997). https://doi.org/10.1007/BF01252226
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DOI: https://doi.org/10.1007/BF01252226