The Orlicz-Minkowski problem for polytopes

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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 and 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.
[2]	J. Boroszky, P. Hegedus and G. Zhu, On the discrete logarithmic Minkowski problem, IMRS., 2016 (2016), 1807-1838. doi: 10.1093/imrn/rnv189.
[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.
[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.
[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.
[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.
[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.
[8]	P. Guan and C. -S. Lin, On the equation $det(u_ij+\delta_ij u) = f$, preprint, 2004.
[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.
[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.
[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.
[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.
[13]	M. -Y. Jiang, The Planar Discrete $L_p$-Minkowski Problem for $p < 1$, preprint, 2014.
[14]	M. -Y. Jiang and C. Wang, The $L_p$-Minkowski Problem for Polytopes with $p < 1$, preprint, 2017.
[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.
[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.
[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.
[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.
[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.
[20]	E. Lutwak, The Brunn-Minkowski-Firey theory, I, Mixed volume and the Minkowski problem, J. Differential Geometry, 38 (1993), 131-150.
[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.
[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.
[23]	A. V. Pogorelov, The Minkowski Multipledimensional Problem, V. H. Winston & Sons, Washington D. C., 1978.
[24]	R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511526282.
[25]	A. Stancu, The discrete $L_0$-Minkowski problem, Adv. in Math., 167 (2002), 160-174. doi: 10.1006/aima.2001.2040.
[26]	C. Wang, Discrete Orlicz-Minkowski problem and Q-curvature equation in Dimension 1, Ph. D thesis, Peking University, 2018.
[27]	G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. in Math., 262 (2014), 909-931. doi: 10.1016/j.aim.2014.06.004.
[28]	G. Zhu, The centroaffine Minkowski problem for polytopes, J. Differential Geometry., 101 (2015), 159-174.
[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.
[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.

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.
[2]	J. Boroszky, P. Hegedus and G. Zhu, On the discrete logarithmic Minkowski problem, IMRS., 2016 (2016), 1807-1838. doi: 10.1093/imrn/rnv189.
[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.
[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.
[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.
[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.
[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.
[8]	P. Guan and C. -S. Lin, On the equation $det(u_ij+\delta_ij u) = f$, preprint, 2004.
[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.
[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.
[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.
[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.
[13]	M. -Y. Jiang, The Planar Discrete $L_p$-Minkowski Problem for $p < 1$, preprint, 2014.
[14]	M. -Y. Jiang and C. Wang, The $L_p$-Minkowski Problem for Polytopes with $p < 1$, preprint, 2017.
[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.
[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.
[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.
[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.
[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.
[20]	E. Lutwak, The Brunn-Minkowski-Firey theory, I, Mixed volume and the Minkowski problem, J. Differential Geometry, 38 (1993), 131-150.
[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.
[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.
[23]	A. V. Pogorelov, The Minkowski Multipledimensional Problem, V. H. Winston & Sons, Washington D. C., 1978.
[24]	R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511526282.
[25]	A. Stancu, The discrete $L_0$-Minkowski problem, Adv. in Math., 167 (2002), 160-174. doi: 10.1006/aima.2001.2040.
[26]	C. Wang, Discrete Orlicz-Minkowski problem and Q-curvature equation in Dimension 1, Ph. D thesis, Peking University, 2018.
[27]	G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. in Math., 262 (2014), 909-931. doi: 10.1016/j.aim.2014.06.004.
[28]	G. Zhu, The centroaffine Minkowski problem for polytopes, J. Differential Geometry., 101 (2015), 159-174.
[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.
[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.

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