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The Orlicz-Minkowski problem for polytopes
1. | LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China |
2. | The Affiliated High School of Peking University, Beijing, 100080, China |
The Orlicz-Minkowski problem for polytopes is studied, and some existence results are established by the variational method.
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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