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THE ISOPERIMETRIC QUOTIENT OF A CONVEX BODY DECREASES MONOTONICALLY UNDER THE EIKONAL ABRASION MODEL

Part of: Equations of mathematical physics and other areas of application General convexity

Published online by Cambridge University Press:  30 August 2018

Gábor Domokos  and
Zsolt Lángi
Gábor Domokos
Affiliation:
MTA-BME Morphodynamics Research Group and Department of Mechanics, Materials and Structures, Budapest University of Technology, Műegyetem rakpart 1-3., Budapest 1111, Hungary email domokos@iit.bme.hu
Zsolt Lángi
Affiliation:
MTA-BME Morphodynamics Research Group and Department of Geometry, Budapest University of Technology, Egry József utca 1., Budapest 1111, Hungary email zlangi @math.bme.hu
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Abstract

We show that under the Eikonal abrasion model, prescribing uniform normal speed in the direction of the inward surface normal, the isoperimetric quotient of a convex shape is decreasing monotonically.

MSC classification

Primary: 35Q85: PDEs in connection with astronomy and astrophysics
Secondary: 35Q86: PDEs in connection with geophysics 52A39: Mixed volumes and related topics
Type
Research Article
Information
Mathematika , Volume 65 , Issue 1 , 2019 , pp. 119 - 129
Copyright
Copyright © University College London 2018 

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References

Arnold, V. I., Lectures on Partial Differential Equations, Springer (2003).Google Scholar
Bloore, F., The shape of pebbles. J. Math. Geol. 9 1977, 113–122.Google Scholar
Blott, S. J. and Pye, K., Particle shape: a review and new methods of characterization and classification. J. Sedimentology 55 2008, 331–363.Google Scholar
Damon, J., Local Morse theory for solutions to the heat eqiation and Gaussian blurring. J. Differential Equations 115 1995, 368–401.Google Scholar
Domokos, G., Monotonicity of spatial critical points evolving under curvature-driven flows. J. Nonlinear Sci. 25(2) 2014, 247–275.Google Scholar
Domokos, G. and Gibbons, G. W., The evolution of pebbles size and shape in space and time. Proc. R. Soc. Lond. A 468 2012, 3059–3079.Google Scholar
Domokos, G. and Lángi, Z., The evolution of geological shape descriptors under distance-driven flows. Math. Geosci. 51(3) 2018, 337–363.Google Scholar
Domokos, G., Sipos, A. Á., Szabó, G. M. and Várkonyi, P. L., Formation of sharp edges and planar areas of asteroids by polyhedral abrasion. Astrophys. J. Lett. 699 2009, L13–L16, doi:10.1088/0004-637X/699/1/L13.Google Scholar
Domokos, G., Sipos, A. Á. and Várkonyi, P., Continuous and discrete models of abrasion processes. Period. Polytech. Archit. 40(1) 2009, 3–8.Google Scholar
Domokos, G., Sipos, A. Á., Várkonyi, P. L. and Szabó, Gy., Explaining the elongated shape of ’Oumuamua by the Eikonal abrasion model. Res. Notes AAS 1 2017, 50, http://stacks.iop.org/2515-5172/1/i=1/a=50.Google Scholar
Dudov, S. I. and Meshcheryakova, E. A., On asphericity of convex bodies. Russian Math. (Iz. VUZ) 59(2) 2015, 36–47.Google Scholar
Firey, W. J., Shapes of worn stones. Mathematika 21 1974, 1–11.Google Scholar
Gage, M. E., An isoperimetric inequality with applications to curve shortening. Duke Math. J. 50 1983, 1225–1229.Google Scholar
Grayson, M. A., The heat equation shrinks embedded plane curves to round points. J. Differential Geom. 26 1987, 285–314.Google Scholar
Gruber, P. M., Convex and Discrete Geometry (A Series of Comprehensive Studies in Mathematics 336 ), Springer (New York, NY, 2007).Google Scholar
Hamilton, R., Isoperimetric estimates for the curve shrinking flow in the plane. Ann. of Math. Stud. 137 1995, 201–222.Google Scholar
Huisken, G., A distance comparison principle for evolving curves. Asian J. Math. 2 1998, 127–134.Google Scholar
Kardar, M., Paris, G. and Zhang, Y. C., Dynamical scaling of growing interfaces. Phys. Rev. Lett. 56 1986, 889–892.Google Scholar
Knill, O., On nonconvex caustics of convex billiards. Elem. Math. 53 1998, 89–106.Google Scholar
Koenderink, J. J., The structure of images. Biol. Cybernet. 50 1984, 363–370.Google Scholar
Larson, S., A bound for the perimeter of inner parallel bodies. J. Funct. Anal. 271 2016, 610–619.Google Scholar
Lindelöf, L., Propriétés générales des polyèdres qui, sous une étendue superficielle donnée, renferment le plus grand volume. Bull. Acad. Imp. Sci. Saint-Pétersbourg 14 1869, 257–269 (extract in Math. Ann. 2 (1870), 150–159).Google Scholar
Lu, C., Cao, Y. and Mumford, D., Surface evolution under curvature flows. J. Visual Commun. Image Represent. 13 2002, 65–81.Google Scholar
Marsilli, M., Maritan, A., Toigo, F. and Banavar, J. B., Stochastic growth equations and reparameterization invariance. Rev. Modern Phys. 68 1996, 963–983.Google Scholar
Pisanski, T., Kaufman, M., Bokal, D., Kirby, E. C. and Graovac, A., Isoperimetric quotient for fullerenes and other polyhedral cages. J. Chem. Inf. Comput. Sci. 37 1997, 1028–1032.Google Scholar
Schneider, R., Convex Bodies: The Brunn–Minkowski Theory, 2nd edn., Cambridge Unversity Press (Cambridge, 2014).Google Scholar
Stachó, L. L., On curvature measures. Acta Sci. Math. (Szeged) 41 1979, 191–207.Google Scholar
Várkonyi, P. L., Laity, E. J. and Domokos, G., Quantitative modeling of facet development in ventifacts by wind abrasion. Aeolian Res. 20 2016, 25–33.Google Scholar