RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN ${L_p}$ SURFACE AREA MEASURES

RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN ${L_p}$ SURFACE AREA MEASURES

RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN ${L_p}$ SURFACE AREA MEASURES

RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN ${L_p}$ SURFACE AREA MEASURES

RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN ${L_p}$ SURFACE AREA MEASURES

RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN Lp{L_p} SURFACE AREA MEASURES

RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN Lp {L_p} SURFACE AREA MEASURES

RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN ${L_p}$ SURFACE AREA MEASURES

RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN ${L_p}$ SURFACE AREA MEASURES

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doi:10.1007/s10473-025-0110-1

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (1): 126-142.doi: 10.1007/s10473-025-0110-1

Juewei Hu, Gangsong Leng

Department of Mathematics, Shanghai University, Shanghai 200444, China

收稿日期:2024-08-30 发布日期:2025-02-06
作者简介:Juewei Hu, E-mail,: syxx40227@163.com; Gangsong Leng, E-mail,: lenggangsong@163.com
基金资助:
NSFC (12171304).

Juewei Hu, Gangsong Leng

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received:2024-08-30 Published:2025-02-06
About author:Juewei Hu, E-mail,: syxx40227@163.com; Gangsong Leng, E-mail,: lenggangsong@163.com
Supported by:
NSFC (12171304).

摘要： We build a computer program to reconstruct convex bodies using even $L_p$ surface area measures for $p\geq 1.$ Firstly, we transform the minimization problem $\mathcal{P}_1$ , which is equivalent to solving the even $L_p$ Minkowski problem, into a convex optimization problem $\mathcal{P}_4$ with a finite number of constraints. This transformation makes it suitable for computational resolution. Then, we prove that the approximate solutions obtained by solving the problem $\mathcal{P}_4$ converge to the theoretical solution when $N$ and $k$ are sufficiently large. Finally, based on the convex optimization problem $\mathcal{P}_4$ , we provide an algorithm for reconstructing convex bodies from even $L_p$ surface area measures, and present several examples implemented using MATLAB.

关键词: reconstruction problem, even $L_p$ surface area measures, spherical harmonic

Abstract: We build a computer program to reconstruct convex bodies using even $L_p$ surface area measures for $p\geq 1.$ Firstly, we transform the minimization problem $\mathcal{P}_1$ , which is equivalent to solving the even $L_p$ Minkowski problem, into a convex optimization problem $\mathcal{P}_4$ with a finite number of constraints. This transformation makes it suitable for computational resolution. Then, we prove that the approximate solutions obtained by solving the problem $\mathcal{P}_4$ converge to the theoretical solution when $N$ and $k$ are sufficiently large. Finally, based on the convex optimization problem $\mathcal{P}_4$ , we provide an algorithm for reconstructing convex bodies from even $L_p$ surface area measures, and present several examples implemented using MATLAB.

Key words: reconstruction problem, even $L_p$ surface area measures, spherical harmonic

中图分类号:

52A40

Juewei Hu, Gangsong Leng. RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN ${L_p}$ SURFACE AREA MEASURES[J]. 数学物理学报(英文版), 2025, 45(1): 126-142.

Juewei Hu, Gangsong Leng. RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN ${L_p}$ SURFACE AREA MEASURES[J]. Acta mathematica scientia,Series B, 2025, 45(1): 126-142.

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[1]	陈守银, 李落清. REALIZATION OF PEETRE K-MODULUS ON THE SPHERE[J]. 数学物理学报(英文版), 1998, 18(4): 427-434.
[2]	李落清, 杨汝月. ALMOST EVERYWHERE CONVERGENCE AND APPROXIMATION OF SPHERICAL HARMONIC EXPANSIONS[J]. 数学物理学报(英文版), 1995, 15(S1): 109-122.

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