American Institute of Mathematical Sciences

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June  2021, 14(6): 1917-1930. doi: 10.3934/dcdss.2021043

The Orlicz-Minkowski problem for polytopes

Meiyue Jiang 1,, and  Chu Wang 2,

1. 

LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China
2. 

The Affiliated High School of Peking University, Beijing, 100080, China

* Corresponding author: Meiyue Jiang

Received  December 2020 Revised  March 2021 Published  June 2021 Early access  April 2021

Fund Project: The first author is supported by NSFC grants 11371038, 11431001, 11971026

The Orlicz-Minkowski problem for polytopes is studied, and some existence results are established by the variational method.

Keywords: Orlicz-Minkowski Problem, Polytopes.
Mathematics Subject Classification: Primary: 52A40; Secondary: 52A20.
Citation: Meiyue Jiang, Chu Wang. The Orlicz-Minkowski problem for polytopes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1917-1930. doi: 10.3934/dcdss.2021043
References:
[1]

J. Ai, K.-S. Chou and J. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var., 13 (2001), 311-337.  doi: 10.1007/s005260000075.  Google Scholar

[2]

J. Boroszky, P. Hegedus and G. Zhu, On the discrete logarithmic Minkowski problem, IMRS., 2016 (2016), 1807-1838.  doi: 10.1093/imrn/rnv189.  Google Scholar

[3]

J. Boroszky, E. Lutwak, D. Yang and G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc., 26 (2013), 831-852.  doi: 10.1090/S0894-0347-2012-00741-3.  Google Scholar

[4]

W. Chen, $L_p$-Minkowski problem with not necessarily positive data, Adv. in Math., 201 (2006), 77-89.  doi: 10.1016/j.aim.2004.11.007.  Google Scholar

[5]

S.-Y. Cheng and S.-T. Yau, On the regularity of the solution of the n-dimensional Minkowski problem,, Comm. Pure Appl. Math., 29 (1976), 495-516.  doi: 10.1002/cpa.3160290504.  Google Scholar

[6]

K.-S. Chou and X.-J. Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine gemotetry,, Adv. in Math., 205 (2006), 33-83.  doi: 10.1016/j.aim.2005.07.004.  Google Scholar

[7]

J. Dou and M. Zhu, The two-dimensional $L_p$-Minkowski problem and nonlinear equations with negative exponents, Adv. in Math., 230 (2012), 1209-1221.  doi: 10.1016/j.aim.2012.02.027.  Google Scholar

[8]

P. Guan and C. -S. Lin, On the equation $det(u_ij+\delta_ij u) = f$, preprint, 2004. Google Scholar

[9]

C. Haberl, E. Lutwak, D. Yang and G. Zhang, The Even Orlicz-Minkowski problem, Adv. in Math., 224 (2010), 2485-2510.  doi: 10.1016/j.aim.2010.02.006.  Google Scholar

[10]

Q. Huang and B. He, On the Orlicz-Minkowski problem for polytopes, Discrete and Comput. Geom., 48 (2012), 281-297.  doi: 10.1007/s00454-012-9434-4.  Google Scholar

[11]

D. Hug, E. Lutwak, D. Yang and G. Zhang, On the $L_p$-Minkowski problem for polytopes,, Discrete and Comput. Geom., 33 (2005), 699-715.  doi: 10.1007/s00454-004-1149-8.  Google Scholar

[12]

M.-Y. Jiang, Remarks on the 2-dimensional $L_p$-Minkowski problem,, Adv. Nonlinear Studies, 10 (2010), 297-313.  doi: 10.1515/ans-2010-0204.  Google Scholar

[13]

M. -Y. Jiang, The Planar Discrete $L_p$-Minkowski Problem for $p < 1$, preprint, 2014. Google Scholar

[14]

M. -Y. Jiang and C. Wang, The $L_p$-Minkowski Problem for Polytopes with $p < 1$, preprint, 2017. Google Scholar

[15]

M.-Y. Jiang, L. Wang and J. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem,, Calc. Var. Partial Differential Equations, 41 (2011), 535-565.  doi: 10.1007/s00526-010-0375-6.  Google Scholar

[16]

M.-Y. Jiang and J. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem II,, Discrete and Continuous Dynam. Systems, 36 (2016), 785-803.  doi: 10.3934/dcds.2016.36.785.  Google Scholar

[17]

F. John, Polar correspondence with respect to a convex region, Duke Math. J., 3 (1937), 355-369.  doi: 10.1215/S0012-7094-37-00327-2.  Google Scholar

[18]

H. Lewy, On the differential geometry in the large, I. Minkowski problem, Trans. Amer. Math. Soc., 43 (1938), 258-270.  doi: 10.2307/1990042.  Google Scholar

[19]

J. Lu and X.-J. Wang, Rotational symmetric solutions to the $L_p$-Minkowski problem,, J. Differential Equations, 254 (2013), 983-1005.  doi: 10.1016/j.jde.2012.10.008.  Google Scholar

[20]

E. Lutwak, The Brunn-Minkowski-Firey theory, I, Mixed volume and the Minkowski problem, J. Differential Geometry, 38 (1993), 131-150.   Google Scholar

[21]

E. Lutwak, D. Yang and G. Zhang, On the $L_p$-Minkowski problem,, Trans. Amer. Math. Soc., 356 (2004), 4359-4370.  doi: 10.1090/S0002-9947-03-03403-2.  Google Scholar

[22]

L. Nirenberg, The Weyl and Minkowski problems in the differential geometry in the large,, Comm. Pure Appl. Math., 6 (1953), 337-394.  doi: 10.1002/cpa.3160060303.  Google Scholar

[23]

A. V. Pogorelov, The Minkowski Multipledimensional Problem, V. H. Winston & Sons, Washington D. C., 1978.  Google Scholar

[24] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511526282.  Google Scholar
[25]

A. Stancu, The discrete $L_0$-Minkowski problem, Adv. in Math., 167 (2002), 160-174.  doi: 10.1006/aima.2001.2040.  Google Scholar

[26]

C. Wang, Discrete Orlicz-Minkowski problem and Q-curvature equation in Dimension 1, Ph. D thesis, Peking University, 2018. Google Scholar

[27]

G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. in Math., 262 (2014), 909-931.  doi: 10.1016/j.aim.2014.06.004.  Google Scholar

[28]

G. Zhu, The centroaffine Minkowski problem for polytopes, J. Differential Geometry., 101 (2015), 159-174.   Google Scholar

[29]

G. Zhu, The $L_p$ Minkowski problem for polytopes for $0 < p < 1$, J. Functional Analysis, 269 (2015), 1070-1094.  doi: 10.1016/j.jfa.2015.05.007.  Google Scholar

[30]

G. Zhu, The $L_p$ Minkowski problem for polytopes for $p < 0$, Indiana Univ. Math. J, 66 (2017), 1333-1350.  doi: 10.1512/iumj.2017.66.6110.  Google Scholar

show all references

References:
[1]

J. Ai, K.-S. Chou and J. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var., 13 (2001), 311-337.  doi: 10.1007/s005260000075.  Google Scholar

[2]

J. Boroszky, P. Hegedus and G. Zhu, On the discrete logarithmic Minkowski problem, IMRS., 2016 (2016), 1807-1838.  doi: 10.1093/imrn/rnv189.  Google Scholar

[3]

J. Boroszky, E. Lutwak, D. Yang and G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc., 26 (2013), 831-852.  doi: 10.1090/S0894-0347-2012-00741-3.  Google Scholar

[4]

W. Chen, $L_p$-Minkowski problem with not necessarily positive data, Adv. in Math., 201 (2006), 77-89.  doi: 10.1016/j.aim.2004.11.007.  Google Scholar

[5]

S.-Y. Cheng and S.-T. Yau, On the regularity of the solution of the n-dimensional Minkowski problem,, Comm. Pure Appl. Math., 29 (1976), 495-516.  doi: 10.1002/cpa.3160290504.  Google Scholar

[6]

K.-S. Chou and X.-J. Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine gemotetry,, Adv. in Math., 205 (2006), 33-83.  doi: 10.1016/j.aim.2005.07.004.  Google Scholar

[7]

J. Dou and M. Zhu, The two-dimensional $L_p$-Minkowski problem and nonlinear equations with negative exponents, Adv. in Math., 230 (2012), 1209-1221.  doi: 10.1016/j.aim.2012.02.027.  Google Scholar

[8]

P. Guan and C. -S. Lin, On the equation $det(u_ij+\delta_ij u) = f$, preprint, 2004. Google Scholar

[9]

C. Haberl, E. Lutwak, D. Yang and G. Zhang, The Even Orlicz-Minkowski problem, Adv. in Math., 224 (2010), 2485-2510.  doi: 10.1016/j.aim.2010.02.006.  Google Scholar

[10]

Q. Huang and B. He, On the Orlicz-Minkowski problem for polytopes, Discrete and Comput. Geom., 48 (2012), 281-297.  doi: 10.1007/s00454-012-9434-4.  Google Scholar

[11]

D. Hug, E. Lutwak, D. Yang and G. Zhang, On the $L_p$-Minkowski problem for polytopes,, Discrete and Comput. Geom., 33 (2005), 699-715.  doi: 10.1007/s00454-004-1149-8.  Google Scholar

[12]

M.-Y. Jiang, Remarks on the 2-dimensional $L_p$-Minkowski problem,, Adv. Nonlinear Studies, 10 (2010), 297-313.  doi: 10.1515/ans-2010-0204.  Google Scholar

[13]

M. -Y. Jiang, The Planar Discrete $L_p$-Minkowski Problem for $p < 1$, preprint, 2014. Google Scholar

[14]

M. -Y. Jiang and C. Wang, The $L_p$-Minkowski Problem for Polytopes with $p < 1$, preprint, 2017. Google Scholar

[15]

M.-Y. Jiang, L. Wang and J. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem,, Calc. Var. Partial Differential Equations, 41 (2011), 535-565.  doi: 10.1007/s00526-010-0375-6.  Google Scholar

[16]

M.-Y. Jiang and J. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem II,, Discrete and Continuous Dynam. Systems, 36 (2016), 785-803.  doi: 10.3934/dcds.2016.36.785.  Google Scholar

[17]

F. John, Polar correspondence with respect to a convex region, Duke Math. J., 3 (1937), 355-369.  doi: 10.1215/S0012-7094-37-00327-2.  Google Scholar

[18]

H. Lewy, On the differential geometry in the large, I. Minkowski problem, Trans. Amer. Math. Soc., 43 (1938), 258-270.  doi: 10.2307/1990042.  Google Scholar

[19]

J. Lu and X.-J. Wang, Rotational symmetric solutions to the $L_p$-Minkowski problem,, J. Differential Equations, 254 (2013), 983-1005.  doi: 10.1016/j.jde.2012.10.008.  Google Scholar

[20]

E. Lutwak, The Brunn-Minkowski-Firey theory, I, Mixed volume and the Minkowski problem, J. Differential Geometry, 38 (1993), 131-150.   Google Scholar

[21]

E. Lutwak, D. Yang and G. Zhang, On the $L_p$-Minkowski problem,, Trans. Amer. Math. Soc., 356 (2004), 4359-4370.  doi: 10.1090/S0002-9947-03-03403-2.  Google Scholar

[22]

L. Nirenberg, The Weyl and Minkowski problems in the differential geometry in the large,, Comm. Pure Appl. Math., 6 (1953), 337-394.  doi: 10.1002/cpa.3160060303.  Google Scholar

[23]

A. V. Pogorelov, The Minkowski Multipledimensional Problem, V. H. Winston & Sons, Washington D. C., 1978.  Google Scholar

[24] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511526282.  Google Scholar
[25]

A. Stancu, The discrete $L_0$-Minkowski problem, Adv. in Math., 167 (2002), 160-174.  doi: 10.1006/aima.2001.2040.  Google Scholar

[26]

C. Wang, Discrete Orlicz-Minkowski problem and Q-curvature equation in Dimension 1, Ph. D thesis, Peking University, 2018. Google Scholar

[27]

G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. in Math., 262 (2014), 909-931.  doi: 10.1016/j.aim.2014.06.004.  Google Scholar

[28]

G. Zhu, The centroaffine Minkowski problem for polytopes, J. Differential Geometry., 101 (2015), 159-174.   Google Scholar

[29]

G. Zhu, The $L_p$ Minkowski problem for polytopes for $0 < p < 1$, J. Functional Analysis, 269 (2015), 1070-1094.  doi: 10.1016/j.jfa.2015.05.007.  Google Scholar

[30]

G. Zhu, The $L_p$ Minkowski problem for polytopes for $p < 0$, Indiana Univ. Math. J, 66 (2017), 1333-1350.  doi: 10.1512/iumj.2017.66.6110.  Google Scholar

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