American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2021037

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

Yuxin Tan 1, and  Yijing Sun 2,,

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  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} $
.
Keywords: Orlicz Minkowski problem, variational method.
Mathematics Subject Classification: Primary: 52A40.
Citation: Yuxin Tan, Yijing Sun. The Orlicz Minkowski problem involving $ 0 < p < 1 $: From one constant to an infinite interval. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021037
References:
[1]

J. Ai, K.-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.  Google Scholar

[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. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[40]

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

[41]

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

[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.  Google Scholar

[43]

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

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

[46]

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

[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.  Google Scholar

[48]

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

[49]

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

[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.  Google Scholar

[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.  Google Scholar

[52]

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

[53]

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

[54]

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

[55]

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

[56]

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

[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.  Google Scholar

[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.  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. Partial Differential Equations, 13 (2001), 311-337.  doi: 10.1007/s005260000075.  Google Scholar

[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. Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[40]

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

[41]

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

[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.  Google Scholar

[43]

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

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

[46]

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

[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.  Google Scholar

[48]

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

[49]

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

[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.  Google Scholar

[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.  Google Scholar

[52]

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

[53]

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

[54]

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

[55]

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

[56]

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

[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.  Google Scholar

[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.  Google Scholar

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