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.
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
.
DOI: 10.1007/s10473-025-0110-1
[1] Alexandroff A. Existence and uniqueness of a convex surface with a given integral curvature. C R (Doklady) Acad Sci URSS (N S), 1942, 35: 131-134
[2] Bianchi G, Böröczky K J, Colesanti A, Yang D. The $L_p$-Minkowski problem for $- n＜p＜1.$ Advances in Mathematics, 2019, 341: 493-535
[3] Böröczky K J, Lutwak E, Yang D, Zhang G Y. The logarithmic minkowski problem. J Amer Math Soc, 2013, 26(3): 831-852
[4] Luis~A, Caffarelli. Interior $W^{2,p}$ estimates for solutions of the monge-ampère equation. Ann of Math, 1990, 131(1): 135-150
[5] Chen S B, Li Q R, Zhu G X. On the $L_p$ Monge-Ampère equation. J Differential Equations, 2017, 263(8): 4997-5011
[6] Cheng S Y, Yau S T. On the regularity of the solution of the $n$-dimensional Minkowski problem. Comm Pure Appl Math, 1976, 29(5): 495-516
[7] Chou K S, Wang X J. The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv Math, 2006, 205(1): 33-83
[8] Fenchel W, Jessen B. Mengenfunktionen Und Konvexe Körper. Kobenhavn: Levin & Munksgaard, 1938
[9] Gardner R J, Kiderlen M, Milanfar P. Convergence of algorithms for reconstructing convex bodies and directional measures. Ann Statist, 2006, 34(3): 1331-1374
[10] Hug D, Lutwak E, Yang D, Zhang G Y. On the $L_p$ Minkowski problem for polytopes. Discrete Comput Geom, 2005, 33(4): 699-715
[11] Jian H Y, Lu J, Wang X J. Nonuniqueness of solutions to the $L_p$-Minkowski problem. Adv Math, 2015, 281: 845-856
[12] Lamberg L, Kaasalainen M. Numerical solution of the minkowski problem. J Comput Appl Math, 2001, 137(2): 213-227
[13] Leng G S, Liu C, Xi D M. Reconstruction problems of convex bodies from surface area measures and lightness functions. Canadian Journal of Mathematics, 2023, 75(5): 1685-1710
[14] Li Q R, Sheng W M, Ye D P, Yi C H. A flow approach to the musielak-orlicz-gauss image problem. Adv Math, 2022, 403: Art 108379
[15] Little J. An iterative method for reconstructing convex polyhedra from extended gaussian images// Proceedings of the Third AAAI Conference on Artificial Intelligence, 1983: 247-250.
[16] Lutwak E. The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski problem. J Differential Geom, 1993, 38(1): 131-150
[17] Minkowski H. Allgemeine Lehrsätze über die convexen Polyeder. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1897, 1897: 198-220
[18] Nirenberg L. The Weyl and Minkowski problems in differential geometry in the large. Comm Pure Appl Math, 1953, 6: 337-394
[19] Pogorelov A V. A regular solution of the $n$-dimensional Minkowski problem. Soviet Math Dokl, 1971, 12: 1192-1196
[20] Prince J L, Willsky A S. Reconstructing convex sets from support line measurements. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1990, 12(4): 377-389
[21] Seeley R T. Spherical harmonics. Amer Math Monthly, 1966, 73: 115-121
