# On the uniqueness of solutions to the isotropic $L_{p}$ dual Minkowski
problem

Yingxiang Hu, Mohammad N. Ivaki<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We prove that the unit sphere is the only smooth, strictly convex solution to
the isotropic $L_{p}$ dual Minkowski problem

$\displaystyle h^{p-1}|Dh|^{n+1-q}\mathcal{K}=1,$

provided $(p,q)\in(-n-1,-1]\times[n,n+1)$.

##  1\. Introduction

An important question in convex geometry is the uniqueness or the non-
uniqueness of the origin-centred spheres as solutions to the isotropic $L_{p}$
dual Minkowski problem

(1.1) $\displaystyle h^{p-1}|Dh|^{n+1-q}\mathcal{K}=c,\quad c\in(0,\infty).$

The $L_{p}$ dual Minkowski problem was first introduced by Lutwak, Yang and
Zhang [LYZ18], acting as a bridge which connects the $L_{p}$-Minkowski problem
to the dual Minkowski problem. The former, the $L_{p}$-Minkowski problem, was
introduced by Lutwak in his influential paper [Lut93] three decades ago, and
since then has been extensively investigated; e.g., [LO95, LYZ04, Sta02,
Sta03, CW06, BLYZ12, BLYZ13, JLW15, Zh15, HLW16, BIS19, BBCY19, Li19, CHLL20,
Mil21, KM22, IM23b] and [Sch14]. The latter, the dual Minkowski problem, was
proposed recently by Huang et. al in [HLYZ16], and further studied in [Zh17,
Zh18, BHP18, CL18, HP18, HJ19, LSW20]. There has been significant progress on
the $L_{p}$ dual Minkowski problem after the paper [LYZ18] such as [HZ18,
BF19, CHZ19, CL21, LLL22]; however, the complete answer to the uniqueness and
non-uniqueness question, as stated above, has been elusive in the most
interesting case: without the _origin-symmetry_ assumption.

Here are the known uniqueness and non-uniqueness results for the isotropic
$L_{p}$ dual Minkowski problem:

  * •

[BCD17], uniqueness of solutions for $-(n+1)\leq p<1$ and $q=n+1$ (see also
[And99, And03, Sar22]);

  * •

[HZ18], uniqueness of solutions for $p>q$;

  * •

[CHZ19], uniqueness of origin-symmetric solutions for

$-(n+1)\leq p<q\leq\min\\{n+1,n+1+p\\};$

  * •

[CL21], uniqueness of solutions for $1<p<q\leq n+1$, or $-(n+1)\leq p<q<-1$,
or the uniqueness of solutions up to rescaling for $p=q$;

  * •

[LW22], complete classification for $n=1$;

  * •

[CCL21], non-uniqueness of solutions under any of the following assumptions:

    1. (i)

$q-2(n+1)>p\geq 0$;

    2. (ii)

$q>0$ and $-q^{\ast}<p<\min\\{0,q-2n-2\\}$, where

$q^{\ast}:=\left\\{\begin{aligned} &\frac{q}{q-n},\quad\text{if $q\geq
n+1$}\\\ &\frac{nq}{q-1},\quad\text{if $1<q<n+1$}\\\ &+\infty,\quad\text{if
$0<q\leq 1;$}\end{aligned}\right.$

    3. (iii)

$p+2(n+1)<q\leq 0$.

In the recent work [IM23a], employing the local Brunn-Minkowski inequality,
the following uniqueness was proved.

######  Theorem.

Let $n\geq 2$ and assume $-(n+1)\leq p$ and $q\leq n+1$, with at least one
being strict. Suppose $\mathcal{M}^{n}$ is a smooth, strictly convex, origin-
centred hypersurface such that $h^{p-1}|Dh|^{n+1-q}\mathcal{K}=c$ with $c>0$.
Then $\mathcal{M}^{n}$ is an origin-centred sphere.

Here, we also employ the local Brunn-Minkowski inequality as our main tool to
establish the following uniqueness result.

######  Theorem 1.1.

Let $n\geq 2$. Suppose $\mathcal{M}^{n}$ is a smooth, strictly convex
hypersurface with $h>0$, such that $h^{p-1}|Dh|^{n+1-q}\mathcal{K}=1$. Suppose
either

  1. (1)

$-(n+1)<p\leq-1$ and $n\leq q\leq n+1$,

  2. (2)

or $-(n+1)\leq p\leq-n$ and $1\leq q<n+1$.

Then $\mathcal{M}^{n}$ is the unit sphere.

##  2\. Background

###  2.1. Convex geometry

Let $(\mathbb{R}^{n+1},\delta:=\langle\,,\rangle,D)$ denote the Euclidean
space with its standard inner product and flat connection, and let
$(\mathbb{S}^{n},\bar{g},\bar{\nabla})$ denote the unit sphere equipped with
its standard round metric and Levi-Civita connection.

Suppose $K$ is a smooth, strictly convex body in $\mathbb{R}^{n+1}$ with the
origin in its interior. Write $\mathcal{M}=\mathcal{M}^{n}=\partial K$ for the
boundary of $K$. The Gauss map of $\mathcal{M}$, denoted by $\nu$, takes the
point $p\in\mathcal{M}$ to its unique unit outward normal
$x=\nu(p)\in\mathbb{S}^{n}$. The support function of $K$ is defined by

$\displaystyle h(x):=\max\\{\langle x,y\rangle:\leavevmode\nobreak\ y\in
K\\},\quad x\in\mathbb{S}^{n}.$

The inverse Gauss map $X=\nu^{-1}:\mathbb{S}^{n}\rightarrow\mathcal{M}$ is
given by

$\displaystyle X(x)=Dh(x)=\bar{\nabla}h(x)+h(x)x,\quad x\in\mathbb{S}^{n}.$

The support function can also be expressed as

$\displaystyle h(x)=\langle X(x),x\rangle=\langle\nu^{-1}(x),x\rangle,\quad
x\in\mathbb{S}^{n}.$

The radial function of $K$ is defined by

$\displaystyle r(x):=|X(x)|=(|\bar{\nabla}h(x)|^{2}+h^{2}(x))^{\frac{1}{2}}.$

Moreover, the Gauss curvature of $\mathcal{M}$ is defined by

$\displaystyle\frac{1}{\mathcal{K}(x)}:=\left.\frac{\det(\bar{\nabla}^{2}h+\bar{g}h)}{\det(\bar{g})}\right|_{x},\quad
x\in\mathbb{S}^{n}.$

Note that the matrix
$A[h]:=\bar{\nabla}^{2}h+h\bar{g}=D^{2}h|_{T\mathbb{S}^{n}}$ is positive-
definite. The eigenvalues of the matrix $A[h]$ with respect to the metric
$\bar{g}$, denoted by $\lambda=(\lambda_{1},\ldots,\lambda_{n})$, are the
principal radii of curvature at the point $X(x)\in\mathcal{M}$. Then
$\sigma_{n}=\mathcal{K}^{-1}=\Pi_{i}\lambda_{i}$. The curvature equation (1.1)
can be reformulated as the following Monge-Ampére equation:

(2.1) $\displaystyle h^{1-p}|Dh|^{q-n-1}\det(\bar{\nabla}^{2}h+\bar{g}h)=c.$

The polar body of $K$ is defined by

$\displaystyle K^{\ast}:=\\{y\in\mathbb{R}^{n+1}:\langle x,y\rangle\leq
1\,\leavevmode\nobreak\ \forall x\in K\\}.$

It is well-known that $K^{\ast}$ is also a smooth, strictly convex body
$K^{\ast}$ in $\mathbb{R}^{n+1}$ with the origin in its interior. Moreover,
the following identity holds

(2.2)
$\frac{h^{n+2}(x)(h^{\ast}(x^{\ast}))^{n+2}}{\mathcal{K}(x)\mathcal{K}^{\ast}(x^{\ast})}=1.$

Here $h^{\ast}$ and $\mathcal{K}^{\ast}$ denote respectively the support
function and Gauss curvature of $K^{\ast}$, and $x^{\ast}:=X(x)/|X(x)|$.

Finally, let us introduce the measure $dV:=h\sigma_{n}d\mu$, where $\mu$ is
the spherical Lebesgue measure of the unit sphere $\mathbb{S}^{n}$. Then the
measure $\sigma_{n}d\mu$ is the surface-area measure of $K$, and
$dV=h\sigma_{n}d\mu$ is a constant multiple of the cone-volume measure of $K$.
We refer to [Sch14] for additional background.

###  2.2. Centro-affine geometry

In this section, we recall some basics from centro-affine geometry. For the
related concepts, we refer the reader to [LHSZ15, NS94] and, in particular, to
the excellent paper by Milman [Mil21].

Let $X:\mathbb{S}^{n}\to\mathcal{M}$ be a smooth embedding of $\mathcal{M}$
(which we consider it to be $Dh$ as in the previous section), and consider the
transversal normal field $\xi(x):=X(x)$ (the centro-affine normal). The
transversal vector $\xi$ induces the volume form $V$ (as in the previous
section), a connection $\nabla$, as well as a metric $g^{\xi}$ on
$\mathbb{S}^{n}$ as follows:

$\displaystyle
V(e_{1},\ldots,e_{n})=\det(dX(e_{1}),\ldots,dX(e_{n}),\xi),\quad e_{i}\in
T\mathbb{S}^{n},$ (2.3) $\displaystyle
D_{u}dX(v)=dX(\nabla_{u}v)-g^{\xi}(u,v)\xi,\quad u,v\in T\mathbb{S}^{n}.$

Note that $g^{\xi}$ is symmetric and positive-definite. Moreover, while
$\nabla$ is not the Levi-Civita connection of $g$, it is torsion-free and

(2.4) $\displaystyle\nabla V\equiv 0.$

The conormal field
$\xi^{\ast}:\mathbb{S}^{n}\to(\mathbb{R}^{n+1})^{\ast}\sim\mathbb{R}^{n+1}$ is
the unique smooth vector field to the dual space of $\mathbb{R}^{n+1}$, such
that $\langle\xi^{\ast},dX\rangle=0$ and $\langle\xi,\xi^{\ast}\rangle=1$.
Moreover, $\xi^{\ast}$ is an immersion and transversal to its image, and it
induces a bilinear form and a torsion-free connection on $\mathbb{S}^{n},$

(2.5) $\displaystyle
D_{u}d\xi^{\ast}(v)=d\xi^{\ast}(\nabla_{u}^{\ast}v)-g^{\xi^{\ast}}(u,v)\xi^{\ast},\quad
u,v\in T\mathbb{S}^{n}.$

We furnish all geometric quantities associated with $\xi^{\ast}$ with $\ast$.

It is known that $g^{\xi}=g^{\xi^{\ast}}$ and that the two connections
$\nabla^{\ast}$ and $\nabla$ are conjugate with respect to $g^{\xi}$:

(2.6) $\displaystyle
ug^{\xi}(v_{1},v_{2})=g^{\xi}(\nabla_{u}v_{1},v_{2})+g^{\xi}(v_{1},\nabla^{\ast}_{u}v_{2})\quad
u,v_{1},v_{2}\in T\mathbb{S}^{n}.$

Moreover, by [Mil21, Proposition 4.2] (or taking the inner production of (2.3)
with $\nu$), we find

(2.7) $\displaystyle g^{\xi}=g^{\xi^{\ast}}=\frac{A[h]}{h}:=g.$

For a smooth function $f:\mathbb{S}^{n}\to\mathbb{R}$, the Hessian and
Laplacian with respect to $(\nabla,g)$ are defined as

$\operatorname{Hess}f(u,v)=\nabla df(u,v)=v(uf)-df(\nabla_{v}u)$

and $\Delta f=\operatorname{div}_{g}(\nabla f)=\sum_{i}g(\nabla_{e_{i}}\nabla
f,e_{i}),$ where $\\{e_{i}\\}_{i=1}^{n}$ is a local $g$-orthonormal frame of
$T\mathbb{S}^{n}$.

We write $\operatorname{Hess}^{\ast}$ and $\Delta^{\ast}$ respectively for the
Hessian and Laplacian with respect to $(\nabla^{\ast},g)$. Since
$\nabla,\nabla^{\ast}$ are conjugate, we have

$\displaystyle v(uf)=vg(\nabla f,u)=g(\nabla_{v}\nabla
f,u)+df(\nabla^{\ast}_{v}u).$

Therefore, we obtain

(2.8) $\displaystyle\Delta
f=\operatorname{tr}_{g}\operatorname{Hess}^{\ast}f,\quad\Delta^{\ast}f=\operatorname{tr}_{g}\operatorname{Hess}f.$

By [Mil21, Proposition 4.2], we have

(2.9)
$\displaystyle\operatorname{Hess}^{\ast}f+gf=\frac{1}{h}\left(\bar{\nabla}^{2}(hf)+\bar{g}hf\right)=\frac{A[hf]}{h}.$

Let us define

(2.10) $\displaystyle Q(u,v)=\nabla^{\ast}_{v}u-\nabla_{v}u\quad\forall u,v\in
T\mathbb{S}^{n}.$

Then by [LHSZ15, (6.2)],

(2.11)
$\displaystyle\operatorname{tr}_{g}Q=-\nabla\log\left(\frac{h^{n+2}}{\mathcal{K}}\right).$

In particular, we have

(2.12)
$\displaystyle(\Delta-\Delta^{\ast})f=-\sum_{i}Q(e_{i},e_{i})f=d\log\frac{h^{n+2}}{\mathcal{K}}(\nabla
f).$

We conclude this section by recalling the local Brunn-Minkowski inequality,
reformulated in the language of centro-affine geometry (cf. [Mil21]): Let
$f\in C^{1}(\mathbb{S}^{n})$. Then

(2.13) $\displaystyle n\int f^{2}dV\leq\int|\nabla f|_{g}^{2}dV+n\frac{(\int
fdV)^{2}}{\int dV}.$

The equality holds if and only if for some $w\in\mathbb{R}^{n+1}$

(2.14) $\displaystyle f(x)=\langle\frac{x}{h(x)},w\rangle\quad\forall
x\in\mathbb{S}^{n}.$

Moreover, by [Mil21, (5.9)] we also have

(2.15) $\displaystyle n\int|\nabla f|_{g}^{2}dV\leq\int(\Delta
f)^{2}dV\quad\forall f\in C^{2}(\mathbb{S}^{n}).$

##  3\. Uniqueness

The following identity is at the heart of our approach to employing the local
Brunn-Minkowski inequality.

######  Theorem 3.1.

There holds

(3.1) $\displaystyle\Delta X+nX=h\bar{\nabla}\log\frac{h^{n+2}}{\mathcal{K}}.$

In particular,

(3.2) $\displaystyle n\int XdV=\int
h\bar{\nabla}\log\frac{h^{n+2}}{\mathcal{K}}dV.$

###### Proof.

Let $w\in\mathbb{R}^{n+1}$ be a fixed vector. By the centro-affine Gauss
equation for $\xi=X$ (cf. [Mil21, Section 3.8]), we have

(3.3) $\displaystyle\Delta^{\ast}\langle X,w\rangle+n\langle X,w\rangle=0.$

Now let $\\{v_{i}\\}_{i=1}^{n}$ be a local orthonormal frame of
$T\mathbb{S}^{n}$ that diagonalizes $A[h]$ at $x_{0}$ and
$A[h]|_{x_{0}}(v_{i},v_{j})=\delta_{ij}\lambda_{i}$. Define
$e_{i}=\sqrt{\frac{h}{\lambda_{i}}}v_{i}$, $i=1,\ldots,n$. Then we have
$g|_{x_{0}}(e_{i},e_{j})=\delta_{ij}$. Hence, by (2.12), at $x_{0}$ we have

(3.4) $\displaystyle\Delta\langle X,w\rangle+n\langle X,w\rangle$
$\displaystyle=(\Delta-\Delta^{\ast})\langle X,w\rangle$
$\displaystyle=g(\nabla\log\frac{h^{n+2}}{\mathcal{K}},\nabla\langle
X,w\rangle)$
$\displaystyle=g(\nabla\log\frac{h^{n+2}}{\mathcal{K}},\lambda_{i}\langle
e_{i},w\rangle e_{i})$ $\displaystyle=\lambda_{i}\langle e_{i},w\rangle
d\log\frac{h^{n+2}}{\mathcal{K}}(e_{i})$
$\displaystyle=\langle\bar{\nabla}\log\frac{h^{n+2}}{\mathcal{K}},hw\rangle.$

The second identity follows from integrating (3.1) against $dV$. ∎

######  Lemma 3.2.

Let $0<f\in C^{2}(\mathbb{S}^{n})$. Then

(3.5) $\displaystyle\int
f^{2}\left(\langle\bar{\nabla}\log\frac{h^{n+2}}{\mathcal{K}},hX\rangle-|X|^{2}|\nabla\log
f|_{g}^{2}\right)dV\leq n\frac{|\int fXdV|^{2}}{\int dV}.$

###### Proof.

Let $\\{E_{k}\\}_{k=1}^{n+1}$ be an orthonormal basis of $\mathbb{R}^{n+1}$.
We define

(3.6) $\displaystyle f_{k}=f\langle X,E_{k}\rangle\quad k=1,\ldots,n+1.$

In view of 3.1, we have

$\displaystyle\Delta f_{k}+nf_{k}=$ $\displaystyle
f\langle\bar{\nabla}\log\frac{h^{n+2}}{\mathcal{K}},hE_{k}\rangle+\langle
X,E_{k}\rangle\Delta f+2g(\nabla f,\nabla\langle X,E_{k}\rangle).$

Therefore,

(3.7) $\displaystyle\sum_{k}f_{k}(\Delta f_{k}+nf_{k})=$ $\displaystyle
f^{2}\langle\bar{\nabla}\log\frac{h^{n+2}}{\mathcal{K}},hX\rangle+f|X|^{2}\Delta
f$ (3.8) $\displaystyle+fg(\nabla f,\nabla|X|^{2}).$

Moreover, by integration by parts (cf. (2.4)), there holds

(3.9) $\displaystyle\int|X|^{2}f\Delta f+fg(\nabla
f,\nabla|X|^{2})dV=-\int|X|^{2}|\nabla f|_{g}^{2}dV.$

By the local Brunn-Minkowski inequality (see (2.13)), we have

(3.10) $\displaystyle\sum_{k}\int f_{k}(\Delta f_{k}+nf_{k})dV\leq
n\sum_{k}\frac{\langle\int fXdV,E_{k}\rangle^{2}}{\int dV}.$

Thus the claim follows from (3.7) and (3.9). ∎

######  Lemma 3.3.

Suppose $\varphi:(0,\infty)\to(0,\infty)$ is $C^{1}$-smooth and
$f=\varphi(r)$. Then we have

(3.11) $\displaystyle\int
f^{2}\langle\bar{\nabla}\log\frac{h^{n+2}}{\mathcal{K}}-(r(\log\varphi)^{\prime})^{2}\bar{\nabla}\log
r,hX\rangle dV\leq n\frac{|\int fXdV|^{2}}{\int dV}.$

###### Proof.

Let $\\{v_{i}\\}_{i=1}^{n}$ and $\\{e_{i}\\}_{i=1}^{n}$ be as in the proof of
3.1. We calculate

(3.12) $\displaystyle e_{i}(\log
f)=(\log\varphi)^{\prime}e_{i}r=\frac{(\log\varphi)^{\prime}}{r}\lambda_{i}\langle
e_{i},X\rangle=\frac{(\log\varphi)^{\prime}}{r}\sqrt{h\lambda_{i}}\langle
v_{i},X\rangle,$

and

(3.13) $\displaystyle r^{2}|\nabla\log
f|_{g}^{2}=((\log\varphi)^{\prime})^{2}h\lambda_{i}(v_{i}h)^{2}=(r(\log\varphi)^{\prime})^{2}\langle\bar{\nabla}\log
r,hX\rangle.$

Now the inequality follows from 3.2. ∎

###### Proof of 1.1.

Let $\alpha=q-n-1$. Due to 3.3 with $\varphi(r)=r^{q-n-1}$, and our assumption
$h^{n+2}\mathcal{K}^{-1}=h^{n+1+p}r^{n+1-q},$ we obtain

(3.14) $\displaystyle(n+1+p)\int r^{2\alpha}|\bar{\nabla}h|^{2}dV$
$\displaystyle\leq\leavevmode\nobreak$ $\displaystyle\alpha(\alpha+1)\int
r^{2\alpha}\langle\bar{\nabla}\log r,h\bar{\nabla}h\rangle dV+n\frac{|\int
r^{\alpha}XdV|^{2}}{\int dV}.$

Assuming $\alpha^{2}+\alpha\leq 0$ (i.e. $n\leq q\leq n+1$) we obtain

(3.15) $\displaystyle(n+1+p)\int r^{2\alpha}|\bar{\nabla}h|^{2}dV\leq
n\frac{|\int r^{\alpha}XdV|^{2}}{\int dV}.$

Moreover, by using $\bar{\Delta}x+nx=0,$

(3.16) $\displaystyle\int r^{\alpha}XdV=\int Xh^{p}d\mu=\frac{n+1+p}{n}\int
r^{\alpha}\bar{\nabla}hdV.$

Hence, due to $n+1+p>0$,

(3.17) $\displaystyle\int
r^{2\alpha}|\bar{\nabla}h|^{2}dV\leq\frac{n+1+p}{n}\frac{|\int
r^{\alpha}\bar{\nabla}hdV|}{\int dV}.$

We may rewrite this inequality as

(3.18) $\displaystyle\int\left|r^{\alpha}\bar{\nabla}h-\frac{\int
r^{\alpha}\bar{\nabla}hdV}{\int dV}\right|^{2}dV\leq\frac{p+1}{n}\frac{|\int
r^{\alpha}\bar{\nabla}hdV|^{2}}{\int dV}.$

Thus $h$ is constant, provided $-(n+1)<p\leq-1$ and $n\leq q\leq n+1$.

In view of (2.2), the polar body $K^{\ast}$ satisfies the following isotropic
$L_{-q}$ dual Minkowski problem:

$\displaystyle(h^{\ast})^{-1-q}|Dh^{\ast}|^{n+1+p}\mathcal{K}^{\ast}=1.$

Hence, the uniqueness result also holds when $n\leq-p\leq(n+1)$ and
$-(n+1)<-q\leq-1$. ∎

## Acknowledgment

The first author’s work was supported by the National Key Research and
Development Program of China 2021YFA1001800 and the National Natural Science
Foundation of China 12101027. Both authors were supported by the Austrian
Science Fund (FWF): Project P36545.

## References

  * [And97] B. Andrews, Monotone quantities and unique limits for evolving convex hypersurfaces, Int. Math. Res. Not. IMRN. 20(1997): 1001–1031.
  * [And99] B. Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138(1999): 151–161.
  * [And03] B. Andrews, Classification of limiting shapes for isotropic curve flows, J. Amer. Math. Soc. 16(2003): 443–459.
  * [ACGL20] B. Andrews, B. Chow, C. Guenther, M. Langford, Extrinsic geometric flows, Amer. Math. Soc. 206, 2020.
  * [BF19] K. Böröczky, F. Fodor, The $L_{p}$ dual Minkowski problem for $p>1$ and $q>0$, J. Differential Equations, 266(2019): 7980–8033.
  * [BHP18] K. Böröczky, M. Henk, H. Pollehn, Subspace concentration of dual curvature measures of symmetric convex bodies, J. Differ. Geom. 109(2018): 411–429.
  * [BBCY19] G. Bianchi, K. J. Böröczky, A. Colesanti, D. Yang, The $L_{p}$-Minkowski problem for $-n<p<1$, Adv. Math. 341(2019): 493–535.
  * [BIS19] P. Bryan, M. N. Ivaki, J. Scheuer, A unified flow approach to smooth, even $L_{p}$-Minkowski problems, Analysis PDE 12(2019): 259–280.
  * [BLYZ12] K. Böröczky, E. Lutwak, D. Yang, G. Zhang, The log-Brunn-Minkowski inequality, Adv. Math. 231(2012): 1974–1997.
  * [BLYZ13] K. Böröczky, E. Lutwak, D. Yang, G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc. 26(2013): 831–852.
  * [BCD17] S. Brendle, K. Choi, P. Daskalopoulos, Asymptotic behavior of flows by powers of the Gaussian curvature, Acta Math. 219(2017): 1–16.
  * [CHZ19] C. Chen, Y. Huang, Y. Zhao, Smooth solutions to the $L_{p}$ dual Minkowski problem, Math. Ann. 373(2019): 953–976.
  * [CCL21] H. Chen, S. Chen, Q.-R. Li, Variations of a class of Monge-Ampére-type functionals and their applications, Analysis PDE 14(2021): 689–716.
  * [CL21] H. Chen, Q.-R. Li, The $L_{p}$ dual Minkowski problem and related parabolic flows, J. Funct. Anal. 281(2021), 108139.
  * [CHLL20] S. Chen, Y. Huang, Q.-R. Li, J. Liu, The $L_{p}$-Brunn–Minkowski inequality for $p<1$, Adv. Math. 368(2020), 107166.
  * [CL18] S. Chen, Q.-R. Li, On the planar dual Minkowski problem, Adv. Math. 333(2018): 87–117.
  * [CW06] K. S. Chou, X.-J. Wang, The $L_{p}$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math. 205(2006): 33–83.
  * [HLW16] Y. He, Q.-R. Li, X.-J. Wang, Multiple solutions of the $L_{p}$-Minkowski problem, Calc. Var. Partial Differential Equations, 55(2016), 117.
  * [HP18] M. Henk, H. Pollehn, Necessary subspace concentration conditions for the even dual Minkowski problem, Adv. Math. 323(2018): 114–141.
  * [HLYZ16] Y. Huang, E. Lutwak, D. Yang, G. Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math. 216(2016): 325–388.
  * [HJ19] Y. Huang, Y. Jiang, Variational characterization for the planar dual Minkowski problem, J. Funct. Anal. 277(2019): 2209–2236.
  * [HZ18] Y. Huang, Y. Zhao, On the $L_{p}$ dual Minkowski problem, Adv. Math. 332(2018): 57–84.
  * [IM23a] M. N. Ivaki, E. Milman, Uniqueness of solutions to a class of isotropic curvature problems, arXiv:2304.12839 (2023).
  * [IM23b] M. N. Ivaki, E. Milman, $L^{p}$-Minkowski problem under curvature pinching, arXiv:2307.16484 (2023).
  * [JLW15] H. Jian, J. Lu, X.-J. Wang, Nonuniqueness of solutions to the $L_{p}$-Minkowski problem, Adv. Math. 281(2015): 845–856.
  * [KM22] A. V. Kolesnikov, E. Milman, Local $L^{p}$-Brunn-Minkowski inequalities for $p<1$, Mem. Amer. Math. Soc. 277(2022): No. 1360.
  * [LHSZ15] A.-M Li, U. Simon, G. Zhao, Z. Hu, Global affine differential geometry of hypersurfaces, de Gruyter, (2015).
  * [Li19] Q.-R. Li, Infinitely many solutions for centro-affine Minkowski problem, Int. Math. Res. Not. IMRN, 18(2019): 5577–5596
  * [LSW20] Q.-R. Li, W.-M. Sheng, X.-J. Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, J. Eur. Math. Soc. (JEMS), 22(2020): 893–923.
  * [LLL22] Q.-R. Li, J. Liu, J. Lu, Nonuniqueness of solutions to the $L_{p}$ dual Minkowski problem, Int. Math. Res. Not. IMRN, 12(2022): 9114–9150.
  * [LW22] H. Li, Y. Wan, Classification of solutions for the planar isotropic $L_{p}$ dual Minkowski problem, arXiv:2209.14630 (2022).
  * [Lut93] E. Lutwak, The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem, J. Differ. Geom. 38(1993): 131–150.
  * [LO95] E. Lutwak, V. Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differ. Geom. 41(1995): 227–246.
  * [LYZ04] E. Lutwak, D. Yang, G. Zhang, On the $L_{p}$-Minkowski problem, Trans. Amer. Math. Soc. 356(2004): 4359–4370.
  * [LYZ18] E. Lutwak, D. Yang, G. Zhang, $L_{p}$ dual curvature measure, Adv. Math. 329(2018): 85–132.
  * [Mil21] E. Milman, Centro-affine differential geometry and the log-Minkowski problem, (to appear in) J. Eur. Math. Soc. arXiv:2104.12408 (2021).
  * [NS94] K. Nomizu, T. Sasaki, _Affine differential geometry_ , Cambridge Univ. Press, (1994).
  * [Sar22] C. Saroglou, On a non-homogeneous version of a problem of Firey, Math. Ann. 382(2022): 1059–1090.
  * [Sch14] R. Schneider, Convex bodies: The Brunn-Minkowski theory, Encyclopedia Math. Appl., Cambridge Univ. Press, New York, second edition (2014).
  * [Sta02] A. Stancu, The discrete planar $L_{0}$-Minkowski problem, Adv. Math. 167(2002): 160–174.
  * [Sta03] A. Stancu, On the number of solutions to the discrete two-dimensional $L_{0}$-Minkowski problem, Adv. Math. 180(2003): 290–323.
  * [Zh17] Y. Zhao, The dual Minkowski problem for negative indices, Calc. Var. Partial Differential Equations, 58(2017), 18.
  * [Zh18] Y. Zhao, Existence of solutions to the even dual Minkowski problem, J. Differ. Geom. 110(2018): 543–572.
  * [Zh15] G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differ. Geom. 101(2015): 159–174.

School of Mathematical Sciences, Beihang University, Beijing 100191, China

Institut für Diskrete Mathematik und Geometrie,
Technische Universität Wien, Wiedner Hauptstraße 8-10,
1040 Wien, Austria,

Institut für Diskrete Mathematik und Geometrie,
Technische Universität Wien, Wiedner Hauptstraße 8-10,
1040 Wien, Austria,