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Reconstruction problems of convex bodies from surface area measures and lightness functions

Part of: Numerical approximation and computational geometry General convexity

Published online by Cambridge University Press:  03 October 2022

Gangsong Leng ,
Chang Liu  and
Dongmeng Xi
Gangsong Leng
Affiliation:
Department of Mathematics, Shanghai University, Shanghai, China e-mail: lenggangsong@163.com liuchangchang316@163.com dongmeng.xi@live.com
Chang Liu
Affiliation:
Department of Mathematics, Shanghai University, Shanghai, China e-mail: lenggangsong@163.com liuchangchang316@163.com dongmeng.xi@live.com
Dongmeng Xi*
Affiliation:
Department of Mathematics, Shanghai University, Shanghai, China e-mail: lenggangsong@163.com liuchangchang316@163.com dongmeng.xi@live.com
*
Dongmeng Xi is the corresponding author. e-mail: dongmeng.xi@live.com
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Abstract

First, we build a computational procedure to reconstruct the convex body from a pre-given surface area measure. Nontrivially, we prove the convergence of this procedure. Then, the sufficient and necessary conditions of a Sobolev binary function to be a lightness function of a convex body are investigated. Finally, we present a computational procedure to compute the curvature function from a pre-given lightness function, and then we reconstruct the convex body from this curvature function by using the first procedure. The convergence is also proved. The main computations in both procedures are simply about the spherical harmonics.

Keywords

Reconstruction problemMinkowski problemreconstructionsurface area measurelightness functionspherical harmonic

MSC classification

Primary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Secondary: 65D15: Algorithms for functional approximation
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 5 , October 2023 , pp. 1685 - 1710
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The work was supported by NSFC Grant No. 12171304, NSFC Grant No. 12071277, and STCSM Grant No. 20JC1412600.

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