June  2021, 14(6): 1983-1994. doi: 10.3934/dcdss.2021037

The Orlicz Minkowski problem involving $ 0 < p < 1 $: From one constant to an infinite interval

1. 

School of Computer Science and Mathematics, Fujian University of Technology, Fuzhou 350118, China

2. 

Department of Mathematics, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Yijing Sun

Received  October 2020 Revised  February 2021 Published  June 2021 Early access  March 2021

Fund Project: The authors are supported by NSFC grants 11971027 and 11771468

In this paper we study the existence of convex bodies for the Orlicz Minkowski problem
$ c\varphi (h_{K})dS(K, \cdot) = d\mu\quad \mbox{on}\, {\mathbb{S}}^{n-1} $
where
$ \mu $
is the given Borel measure on
$ {\mathbb{S}}^{n-1} $
,
$ h_{K} $
is the support function of
$ K $
,
$ S_{K} $
is the surface area measure of
$ K $
, and
$ c $
is a real parameter. We prove that, under assumptions on
$ \varphi $
at
$ {\it infinity} $
, there exists
$ c_{*}>0 $
such that, if
$ c\in [c_{*}, +\infty) $
this problem always has a solution
$ K_{c} $
.
Citation: Yuxin Tan, Yijing Sun. The Orlicz Minkowski problem involving $ 0 < p < 1 $: From one constant to an infinite interval. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1983-1994. doi: 10.3934/dcdss.2021037
References:
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J. AiK.-S. Chou and J. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 13 (2001), 311-337.  doi: 10.1007/s005260000075.

[2]

A. D. Aleksandrov, On the theory of mixed volumes. III. Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies, Mat. Sbornik N.S., 3 (1938), 27-46; On the surface area measure of convex bodies, Mat. Sbornik N.S., 6 (1939), 167-174.

[3]

B. Andrews, Gauss curvature flow: The fate of the rolling stones, Invent. Math., 138 (1999), 151-161.  doi: 10.1007/s002220050344.

[4]

B. Andrews, Classifications of limiting shapes for isotropic curve flows, J. Amer. Math. Soc., 16 (2003), 443-459. doi: 10.1090/S0894-0347-02-00415-0.

[5]

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners Existence Results via the variational approach, Springer-Verlag, London, 2011. doi: 10.1007/978-0-85729-227-8.

[6]

K. J. B$\ddot{o}$r$\ddot{o}$czky, E. Lutwak, D. Yang and G. Zhang, The log-Brunn-Minkowski inequality, Adv. Math., 231 (2012), 1974-1997. doi: 10.1016/j.aim.2012.07.015.

[7]

K. J. B$\ddot{o}$r$\ddot{o}$czky, 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.

[8]

K. J. B$\ddot{o}$r$\ddot{o}$czky and H. T. Trinh, The planar $L_{p}$-Minkowski problem for $0 < p < 1$, Adv. Appl. Math., 87 (2017), 58-81. doi: 10.1016/j.aam.2016.12.007.

[9]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for non-linear second order elliptic equations I. Monge-Amp$\grave{e}$re equation, Comm. Pure Appl. Math., 37 (1984), 369-402. doi: 10.1002/cpa.3160370306.

[10]

E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. J$\ddot{o}$rgens, Michigan Math. J., 5 (1958), 105-126.

[11]

W. Chen, $L_{p}$ Minkowski problem with not necessarily positive data, Adv. Math., 201 (2006), 77-89. doi: 10.1016/j.aim.2004.11.007.

[12]

S. Chen, Q. Li and G. Zhu, On the $L_{p}$ Monge-Amp$\grave{e}$re equation, J. Differential Equations, 263 (2017), 4997-5011. doi: 10.1016/j.jde.2017.06.007.

[13]

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.

[14]

K.-S. Chou and X.-J. Wang, The $L_{p}$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83. doi: 10.1016/j.aim.2005.07.004.

[15]

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

[16]

W. Fenchel and B. Jessen, Mengenfunktionen und konvexe K$\ddot{o}$rper, Danske Vid. Selskab. Mat.-fys. Medd., 16 (1938), 1-31.

[17]

W. J. Firey, Shapes of worn stones, Mathematika, 21 (1974), 1-11. doi: 10.1112/S0025579300005714.

[18]

M. Gage, Evolving planes curves by curvature in relative geometries, Duke Math. J., 72 (1993), 441-466.

[19]

M. Gage and Y. Li, Evolving planes curves by curvature in relative geometries, Duke Math. J., 75 (1994), 79-98.

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

[21]

C. Haberl and F. E. Schuster, Asymmetric affine $L_{p}$ Sobolev inequalities, J. Funct. Anal., 257 (2009), 641-658. doi: 10.1016/j.jfa.2009.04.009.

[22]

C. Haberl, F. E. Schuster and J. Xiao, An asymmetric affine P$\acute{o}$lya-Szeg$\ddot{o}$ principle, Math. Ann., 352 (2012), 517-542. doi: 10.1007/s00208-011-0640-9.

[23]

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

[24]

Y. Huang, J. Liu and L. Xu, On the uniqueness of the $L_{p}$ Minkowski problems: The constant $p$-curvature case in ${\mathbb{R}}^{3}$, Adv. Math., 281 (2015), 906-927. doi: 10.1016/j.aim.2015.02.021.

[25]

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

[26]

M. N. Ivaki, A flow approach to the $L_{-2}$ Minkowski problem, Adv. Appl. Math., 50 (2013), 445-464. doi: 10.1016/j.aam.2012.09.003.

[27]

H. Jian and J. Lu, Existence of solutions to the Orlicz-Minkowski problem, Adv. Math., 344 (2019), 262-288. doi: 10.1016/j.aim.2019.01.004.

[28]

H. Jian, J. Lu and X.-J. Wang, Nonuniqueness of solutions to the $L_{p}$-Minkowski problem, Adv. Math., 281 (2015), 845-856. doi: 10.1016/j.aim.2015.05.010.

[29]

M.-Y. Jiang, Remarks on the 2-dimensional $L_{p}$-Minkowski problem, Adv. Nonlinear Stud., 10 (2010), 297-313. doi: 10.1515/ans-2010-0204.

[30]

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

[31]

M. Ludwig, General affine surface areas, Adv. Math., 224 (2010), 2346-2360. doi: 10.1016/j.aim.2010.02.004.

[32]

M. Ludwig and M. Reitzner, A classification of $ {\rm{SL}} (n)$ invariant valuations, Ann. of Math., 172 (2010), 1219-1267. doi: 10.4007/annals.2010.172.1219.

[33]

E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150. doi: 10.4310/jdg/1214454097.

[34]

E. Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math., 118 (1996), 244-294. doi: 10.1006/aima.1996.0022.

[35]

E. Lutwak and V. Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom., 41 (1995), 227-246. doi: 10.4310/jdg/1214456011.

[36]

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.

[37]

E. Lutwak, D. Yang and G. Zhang, Optimal Sobolev norms and the $L_{p}$-Minkowski problem, Int. Math. Res. Not., (2006), Art. ID 62987, 21 pp. doi: 10.1155/IMRN/2006/62987.

[38]

E. Lutwak, D. Yang and G. Zhang, Orlicz projection bodies, Adv. Math., 223 (2010), 220-242. doi: 10.1016/j.aim.2009.08.002.

[39]

E. Lutwak, D. Yang and G. Zhang, Orlicz centroid bodies, J. Differential Geom., 84 (2010), 365-387. doi: 10.4310/jdg/1274707317.

[40]

M. Meyer and E. Werner, On the $p$-affine surface area, Adv. Math., 152 (2000), 288-313. doi: 10.1006/aima.1999.1902.

[41]

H. Minkowski, Volumen und oberfl$\ddot{a}$che, Math. Ann., 57 (1903), 447-495. doi: 10.1007/BF01445180.

[42]

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

[43]

A. V. Pogorelov, The Minkowski Multidimensional Problem, V. H. Winston Sons, Washington D.C., 1978.

[44] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511526282.
[45]

C. Sch$\ddot{u}$tt and E. Werner, Surface bodies and $p$-affine surface area, Adv. Math., 187 (2004), 98-145. doi: 10.1016/j.aim.2003.07.018.

[46]

A. Stancu, The discrete plannar $L_{0}$-Minkowski problem, Adv. Math., 167 (2002), 160-174. doi: 10.1006/aima.2001.2040.

[47]

A. Stancu, On the number of solutions to the discrete two-dimensional $L_{0}$-Minkowski problem, Adv. Math., 180 (2003), 290-323. doi: 10.1016/S0001-8708(03)00005-7.

[48]

M. Struwe, Variational Methods Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 2008 (Four edition).

[49]

Y. Sun, Existence and uniqueness of solutions to Orlicz Minkowski problems involving $0 <p<1$, Adv. Appl. Math., 101 (2018), 184-214.

[50]

Y. Sun and Y. Long, The planar Orlicz Minkowski problme in the $L^{1}$-sense, Adv. Math., 281 (2015), 1364-1383. doi: 10.1016/j.aim.2015.03.032.

[51]

Y. Sun and D. Zhang, The planar Orlicz Minkowski problem for $p = 0$ without even assumptions, J. Geom. Anal., 29 (2019), 3384-3404. doi: 10.1007/s12220-018-00114-x.

[52]

G. Tzitz$\acute{e}$ica, Sur une nouvelle classe de surfaces, Rend. Circ. Mat. Palermo, 25 (1908), 180-187. 28 (1909), 210-216.

[53]

V. Umanskiy, On the solvability of the two dimensional $L_{p}$-Minkowski problem, Adv. Math., 225 (2010), 3214-3228.

[54]

T. Wang, On the discrete functional $L_{p}$ Minkowski problem, Int. Math. Res. Not. IMRN, (2015), 10563-10585. doi: 10.1093/imrn/rnu256.

[55]

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

[56]

G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom., 101 (2015), 159-174. doi: 10.4310/jdg/1433975485.

[57]

G. Zhu, The $L_{p}$ Minkowski problem for polytopes for $0 < p < 1$, J. Funct. Anal., 269 (2015), 1070-1094. doi: 10.1016/j.jfa.2015.05.007.

[58]

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. AiK.-S. Chou and J. Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 13 (2001), 311-337.  doi: 10.1007/s005260000075.

[2]

A. D. Aleksandrov, On the theory of mixed volumes. III. Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies, Mat. Sbornik N.S., 3 (1938), 27-46; On the surface area measure of convex bodies, Mat. Sbornik N.S., 6 (1939), 167-174.

[3]

B. Andrews, Gauss curvature flow: The fate of the rolling stones, Invent. Math., 138 (1999), 151-161.  doi: 10.1007/s002220050344.

[4]

B. Andrews, Classifications of limiting shapes for isotropic curve flows, J. Amer. Math. Soc., 16 (2003), 443-459. doi: 10.1090/S0894-0347-02-00415-0.

[5]

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners Existence Results via the variational approach, Springer-Verlag, London, 2011. doi: 10.1007/978-0-85729-227-8.

[6]

K. J. B$\ddot{o}$r$\ddot{o}$czky, E. Lutwak, D. Yang and G. Zhang, The log-Brunn-Minkowski inequality, Adv. Math., 231 (2012), 1974-1997. doi: 10.1016/j.aim.2012.07.015.

[7]

K. J. B$\ddot{o}$r$\ddot{o}$czky, 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.

[8]

K. J. B$\ddot{o}$r$\ddot{o}$czky and H. T. Trinh, The planar $L_{p}$-Minkowski problem for $0 < p < 1$, Adv. Appl. Math., 87 (2017), 58-81. doi: 10.1016/j.aam.2016.12.007.

[9]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for non-linear second order elliptic equations I. Monge-Amp$\grave{e}$re equation, Comm. Pure Appl. Math., 37 (1984), 369-402. doi: 10.1002/cpa.3160370306.

[10]

E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. J$\ddot{o}$rgens, Michigan Math. J., 5 (1958), 105-126.

[11]

W. Chen, $L_{p}$ Minkowski problem with not necessarily positive data, Adv. Math., 201 (2006), 77-89. doi: 10.1016/j.aim.2004.11.007.

[12]

S. Chen, Q. Li and G. Zhu, On the $L_{p}$ Monge-Amp$\grave{e}$re equation, J. Differential Equations, 263 (2017), 4997-5011. doi: 10.1016/j.jde.2017.06.007.

[13]

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.

[14]

K.-S. Chou and X.-J. Wang, The $L_{p}$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205 (2006), 33-83. doi: 10.1016/j.aim.2005.07.004.

[15]

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

[16]

W. Fenchel and B. Jessen, Mengenfunktionen und konvexe K$\ddot{o}$rper, Danske Vid. Selskab. Mat.-fys. Medd., 16 (1938), 1-31.

[17]

W. J. Firey, Shapes of worn stones, Mathematika, 21 (1974), 1-11. doi: 10.1112/S0025579300005714.

[18]

M. Gage, Evolving planes curves by curvature in relative geometries, Duke Math. J., 72 (1993), 441-466.

[19]

M. Gage and Y. Li, Evolving planes curves by curvature in relative geometries, Duke Math. J., 75 (1994), 79-98.

[20]

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

[21]

C. Haberl and F. E. Schuster, Asymmetric affine $L_{p}$ Sobolev inequalities, J. Funct. Anal., 257 (2009), 641-658. doi: 10.1016/j.jfa.2009.04.009.

[22]

C. Haberl, F. E. Schuster and J. Xiao, An asymmetric affine P$\acute{o}$lya-Szeg$\ddot{o}$ principle, Math. Ann., 352 (2012), 517-542. doi: 10.1007/s00208-011-0640-9.

[23]

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

[24]

Y. Huang, J. Liu and L. Xu, On the uniqueness of the $L_{p}$ Minkowski problems: The constant $p$-curvature case in ${\mathbb{R}}^{3}$, Adv. Math., 281 (2015), 906-927. doi: 10.1016/j.aim.2015.02.021.

[25]

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

[26]

M. N. Ivaki, A flow approach to the $L_{-2}$ Minkowski problem, Adv. Appl. Math., 50 (2013), 445-464. doi: 10.1016/j.aam.2012.09.003.

[27]

H. Jian and J. Lu, Existence of solutions to the Orlicz-Minkowski problem, Adv. Math., 344 (2019), 262-288. doi: 10.1016/j.aim.2019.01.004.

[28]

H. Jian, J. Lu and X.-J. Wang, Nonuniqueness of solutions to the $L_{p}$-Minkowski problem, Adv. Math., 281 (2015), 845-856. doi: 10.1016/j.aim.2015.05.010.

[29]

M.-Y. Jiang, Remarks on the 2-dimensional $L_{p}$-Minkowski problem, Adv. Nonlinear Stud., 10 (2010), 297-313. doi: 10.1515/ans-2010-0204.

[30]

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

[31]

M. Ludwig, General affine surface areas, Adv. Math., 224 (2010), 2346-2360. doi: 10.1016/j.aim.2010.02.004.

[32]

M. Ludwig and M. Reitzner, A classification of $ {\rm{SL}} (n)$ invariant valuations, Ann. of Math., 172 (2010), 1219-1267. doi: 10.4007/annals.2010.172.1219.

[33]

E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131-150. doi: 10.4310/jdg/1214454097.

[34]

E. Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math., 118 (1996), 244-294. doi: 10.1006/aima.1996.0022.

[35]

E. Lutwak and V. Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom., 41 (1995), 227-246. doi: 10.4310/jdg/1214456011.

[36]

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.

[37]

E. Lutwak, D. Yang and G. Zhang, Optimal Sobolev norms and the $L_{p}$-Minkowski problem, Int. Math. Res. Not., (2006), Art. ID 62987, 21 pp. doi: 10.1155/IMRN/2006/62987.

[38]

E. Lutwak, D. Yang and G. Zhang, Orlicz projection bodies, Adv. Math., 223 (2010), 220-242. doi: 10.1016/j.aim.2009.08.002.

[39]

E. Lutwak, D. Yang and G. Zhang, Orlicz centroid bodies, J. Differential Geom., 84 (2010), 365-387. doi: 10.4310/jdg/1274707317.

[40]

M. Meyer and E. Werner, On the $p$-affine surface area, Adv. Math., 152 (2000), 288-313. doi: 10.1006/aima.1999.1902.

[41]

H. Minkowski, Volumen und oberfl$\ddot{a}$che, Math. Ann., 57 (1903), 447-495. doi: 10.1007/BF01445180.

[42]

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

[43]

A. V. Pogorelov, The Minkowski Multidimensional Problem, V. H. Winston Sons, Washington D.C., 1978.

[44] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511526282.
[45]

C. Sch$\ddot{u}$tt and E. Werner, Surface bodies and $p$-affine surface area, Adv. Math., 187 (2004), 98-145. doi: 10.1016/j.aim.2003.07.018.

[46]

A. Stancu, The discrete plannar $L_{0}$-Minkowski problem, Adv. Math., 167 (2002), 160-174. doi: 10.1006/aima.2001.2040.

[47]

A. Stancu, On the number of solutions to the discrete two-dimensional $L_{0}$-Minkowski problem, Adv. Math., 180 (2003), 290-323. doi: 10.1016/S0001-8708(03)00005-7.

[48]

M. Struwe, Variational Methods Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 2008 (Four edition).

[49]

Y. Sun, Existence and uniqueness of solutions to Orlicz Minkowski problems involving $0 <p<1$, Adv. Appl. Math., 101 (2018), 184-214.

[50]

Y. Sun and Y. Long, The planar Orlicz Minkowski problme in the $L^{1}$-sense, Adv. Math., 281 (2015), 1364-1383. doi: 10.1016/j.aim.2015.03.032.

[51]

Y. Sun and D. Zhang, The planar Orlicz Minkowski problem for $p = 0$ without even assumptions, J. Geom. Anal., 29 (2019), 3384-3404. doi: 10.1007/s12220-018-00114-x.

[52]

G. Tzitz$\acute{e}$ica, Sur une nouvelle classe de surfaces, Rend. Circ. Mat. Palermo, 25 (1908), 180-187. 28 (1909), 210-216.

[53]

V. Umanskiy, On the solvability of the two dimensional $L_{p}$-Minkowski problem, Adv. Math., 225 (2010), 3214-3228.

[54]

T. Wang, On the discrete functional $L_{p}$ Minkowski problem, Int. Math. Res. Not. IMRN, (2015), 10563-10585. doi: 10.1093/imrn/rnu256.

[55]

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

[56]

G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differential Geom., 101 (2015), 159-174. doi: 10.4310/jdg/1433975485.

[57]

G. Zhu, The $L_{p}$ Minkowski problem for polytopes for $0 < p < 1$, J. Funct. Anal., 269 (2015), 1070-1094. doi: 10.1016/j.jfa.2015.05.007.

[58]

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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