On a non-homogeneous version of a problem of Firey

Abstract

We investigate the uniqueness for the Monge–Ampère type equation

$$\begin{aligned} \text {det}(u_{ij}+\delta _{ij}u)_{i,j=1}^{n-1}=G(u),\qquad \text {on} \ \ {\mathbb {S}}^{n-1}, \end{aligned}$$

(1)

where u is the restriction of the support function on the sphere ${\mathbb {S}}^{n-1}$, of a convex body that contains the origin in its interior and $G:(0,\infty )\rightarrow (0,\infty )$ is a continuous function. The problem was initiated by Firey (Mathematika 21(1): 1–11, 1974) who, in the case $G(\theta )=\theta ^{-1}$, asked if $u\equiv 1$ is the unique solution to (1). Recently, Brendle et al. (Acta Mathe 219(1): 1–16, 2017) proved that if $G(\theta )=\theta ^{-p}$, $p>-n-1$, then u has to be constant, providing in particular a complete solution to Firey’s problem. Our primary goal is to obtain uniqueness (or nearly uniqueness) results for (1) for a broader family of functions G. Our approach is very different than the techniques developed in Brendle et al. (2017).

Appendix

This Appendix is devoted to the proof of Proposition 7.2 in full generality. We will need the following fact, which we believe should be well known. Since we were unable to find an explicit reference, we provide a proof here.

Lemma A.1

Let $A>0$. There exists a constant $C=C(A,n)>0$, such that if L is a convex body in ${{\mathbb {R}}}^n$, with absolutely continuous surface area measure with respect to ${{{\mathcal {H}}}^{n-1}}$ and

$$\begin{aligned} \frac{1}{A}\le f_L(v)\le A,\qquad \text {for almost every }v\in {\mathbb {S}}^{n-1}, \end{aligned}$$

(32)

then there exists $a\in {{\mathbb {R}}}^n$, such that

$$\begin{aligned} \frac{1}{C}B_2^n+a\subseteq L\subseteq CB_2^n+a. \end{aligned}$$

Proof

Any constant $C_1,C_2$, etc. that will appear in this proof will denote a positive constant that depends only on A and the dimension n. For an $(n-1)$-dimensional subspace H of ${{\mathbb {R}}}^n$, we write $V_H(\cdot )$ for the volume functional in H.

Recall the well known fact that if $L_1$, $L_2$ are convex bodies with $S_{L_1}\le S_{L_2}$, then it holds $V(L_1)\le V(L_2)$. Using this and (32), we obtain

$$\begin{aligned} \frac{1}{C_1}\le V(L)\le C_1, \end{aligned}$$

(33)

for some $C_1>0$. Moreover, the well known formula

$$\begin{aligned} V_{e^\perp }(L|e^\perp )=\frac{1}{2}\int _{{\mathbb {S}}^{n-1}}|\langle x,e\rangle |dS_L(x),\qquad e\in {\mathbb {S}}^{n-1}\end{aligned}$$

immediately shows that

$$\begin{aligned} \frac{1}{C_2}\le V_{e^\perp }(L|e^\perp )\le C_2, \end{aligned}$$

(34)

for all $e\in {\mathbb {S}}^{n-1}$, where $C_2$ is another positive constant. After a suitable translation, we may assume that the maximal volume ellipsoid $E=tSB_2^n$ contained in L is centered at the origin. Here, S denotes a symmetric positive definite matrix of determinant 1 and $t=(V(E)/V(B_2^n))^{1/n}$. Then, (33) together with the classical theorem of F. John [25], yields

$$\begin{aligned} \frac{1}{C_3}B_2^n\subseteq S^{-1}L\subseteq C_3B_2^n, \end{aligned}$$

for some constant $C_3>0$. Equivalently, we may write

$$\begin{aligned} \frac{1}{C_3}SB_2^n\subseteq L\subseteq C_3SB_2^n. \end{aligned}$$

(35)

Let $\lambda ,\mu$ be the smallest and the largest eigenvalue of S and $e_\lambda ,e_\mu$ be the corresponding eigenvectors respectively. Then, (34) and (35) give

$$\begin{aligned} \frac{1}{C_3^{n-1}}\frac{1}{\lambda }V_{e_\lambda ^\perp }(B_2^n|e_\lambda ^\perp )=V_{e_\lambda ^\perp }((1/C_3)SB_2^n|e_\lambda ^\perp )\le V_{e_\lambda ^\perp }(L|e_\lambda ^\perp )\le C_2 \end{aligned}$$

and

$$\begin{aligned} C_3^{n-1}\frac{1}{\mu }V_{e_\mu ^\perp }(B_2^n|e_\mu ^\perp )=V_{e_\mu ^\perp }(C_3SB_2^n|e_\mu ^\perp )\ge V_{e_\mu ^\perp }(L|e_\mu ^\perp )\ge \frac{1}{C_2}. \end{aligned}$$

Consequently, if $C_4:=C_2C_3^{n-1}$, then $1/C_4\le \lambda \le \mu \le C_4$ and hence, using again (35), we conclude

$$\begin{aligned} \frac{1}{C_3C_4}B_2^n\subseteq L\subseteq C_3C_4 B_2^n. \end{aligned}$$

This completes our proof. $\square$

Proof of Proposition 7.2

We may clearly assume that $o\in \text {int}\,L\cap \text {int}\,M$. First, let us prove Proposition 7.2 without any regularity assumption on the boundaries of L and M, but under the additional assumption that $V{\setminus } \{p\}\subseteq \text {int}\,M$. This, together with the fact that $p\in \text {int}\,L\cap \text {int}\,M$ is clearly equivalent to:

• (i) $\rho _L(v_0)=\rho _M(v_0)$, where $v_0=p/|p|$

• (ii) $\rho _L(v)<\rho _M(v)$, for all $v\in U:=\{x/|x|:x\in V\}$.

Since $f_L$ and $f_M$ are continuous, there are sequences of strictly positive $C^\infty$ functions $\{{\underline{f}}_m\}$ and $\{{\overline{f}}_m\}$, such that ${\underline{f}}_m\rightarrow f_L$ and ${\overline{f}}_m\rightarrow f_M$, uniformly on ${\mathbb {S}}^{n-1}$, while $\int _{{\mathbb {S}}^{n-1}}x{{\underline{f}}_m}d{{{\mathcal {H}}}^{n-1}}=\int _{{\mathbb {S}}^{n-1}}x{{\overline{f}}_m}d{{{\mathcal {H}}}^{n-1}}=o$, for each $m\in {\mathbb {N}}$. By Minkowski’s existence and Uniqueness Theorem, for $m\in {\mathbb {N}}$, there exist uniquely determined up to translation convex bodies $L_m$ and $M_m$, such that $S_{L_m}={\underline{f}}_md{{{\mathcal {H}}}^{n-1}}$ and $S_{M_m}={\overline{f}}_md{{{\mathcal {H}}}^{n-1}}$. The sequences $\{{\underline{f}}_m\}$ and $\{{\overline{f}}_m\}$ are uniformly bounded from above and uniformly away from zero, therefore after suitable translations, as Lemma A.1 shows, the bodies $L_m$ and $M_m$ are contained in and contain a fixed ball. Hence, by taking subsequences, Blaschke’s Selection Theorem shows that we may assume that $L_m\rightarrow {\overline{L}}$ and $M_m\rightarrow {\overline{M}}$ in the Hausdorff metric, for some convex bodies ${\overline{L}}$ and ${\overline{M}}$. However, ${\underline{f}}_m\rightarrow f_L$ and ${\overline{f}}_m\rightarrow f_M$ uniformly and thus weakly on ${\mathbb {S}}^{n-1}$, so ${\overline{L}}$ and ${\overline{M}}$ are translates of L and M respectively. Finally, we may assume that ${\overline{L}}=L$ and ${\overline{M}}=M$. Notice, in addition, that since ${\underline{f}}_m$ and ${\overline{f}}_m$ are positive $C^\infty$ functions, it follows (see [33]) that $L_m$ and $M_m$ are all of class $C_+^2$.

Since $L_m\rightarrow L$ and $M_m\rightarrow M$ (and since $o\in \text {int}\,L_m\cap \text {int}\,M_m$ if m is large enough), we conclude that $\rho _{L_m}/\rho _L\rightarrow 1$ and $\rho _{M_m}/\rho _M\rightarrow 1$, uniformly on ${\mathbb {S}}^{n-1}$. Thus, since $\min _{v\in \text {bd}\,V}(\rho _M(v)/\rho _L(v))>1$, it follows that if m is large enough, then

$$\begin{aligned} \rho _{M_m}(v)>c\rho _{L_m}(v), \qquad \text {for all }v\in \text {bd}\,V, \end{aligned}$$

where $c>1$ is a constant which is independent of m. On the other hand, for $0<\varepsilon <c-1$, it holds

$$\begin{aligned} \rho _{M_m}(v_0)<(1+\varepsilon )\rho _{L_m}(v_0)<c\rho _{L_m}(v_0). \end{aligned}$$

Consequently, there exists $m_0\in {\mathbb {N}}$, such that for any $m\ge m_0$, the minimum

$$\begin{aligned} c_m:=\min _{v\in \text {cl}\,U}\frac{\rho _{M_m}(v)}{\rho _{L_m}(v)} \end{aligned}$$

is attained inside U. This shows that if $m\ge m_0$, there exists $v_m\in U$, such that $\rho _{c_mL_m}(v_m)=c_m\rho _{L_m}(v_m)=\rho _{M_m}(v_m)$, while $\rho _{c_mL_m}(v)\le \rho _{M_m}(v)$, for all $v\in U$. Thus, the triple $(c_mL_m, M_m, v_m)$ satisfies assumptions (i) and (ii) imposed previously and, therefore, satisfies the (weaker) assumptions in the statement of Propostion 7.2. Since $c_mL_m$ and $M_m$ are of class $C^2_+$, we conclude that if $m\ge m_0$, then

$$\begin{aligned} c_m^{n-1}{\underline{f}}_m(\nu _m)=f_{c_mL_m}(\nu _m)\le {\overline{f}}_m(\nu _m), \end{aligned}$$

(36)

where $\nu _m:=\eta _{L_m}(p_m)$ and $p_m:=\rho _{L_m}(v_m)v_m$.

Next, set $W:=\eta _L(V)$ and let $\{\nu _{k_m}\}$ be a subsequence of $\{\nu _m\}$ that converges to some vector $\nu '\in {\mathbb {S}}^{n-1}$. We claim that $\nu '\in \text {cl}\,W=\eta _L(\text {cl}\,V)$. To see this, recall that $p_{k_m}$ is the unique point in $\text {bd}\,L_{k_m}$, such that

$$\begin{aligned} \langle p_{k_m},\nu _{k_m}\rangle =h_{L_{k_m}}(\nu _{k_m}). \end{aligned}$$

By taking a subsequence, we may assume that $p_{k_m}\rightarrow q$, for some point $q\in \text {cl}\,V$. Thus, it holds $\langle q,\nu '\rangle =h_L(\nu ')$ and, therefore, $\nu '=\eta _L(q)\in \eta _L(\text {cl}\,V)=\text {cl}\,W$.

Since ${\underline{f}}_m$ and ${\overline{f}}_m$ converge uniformly on ${\mathbb {S}}^{n-1}$ and since $c_m\rightarrow 1$, we conclude by (36) that

$$\begin{aligned} f_L(\nu ')\le f_M(\nu '). \end{aligned}$$

Notice that, in the argument described above, one can replace V by any open set $V'\subseteq V$. Having this in mind, consider a sequence $\{V'_l\}$ of open sets in $\text {bd}\,L$, all contained in V, such that $V'_l\searrow \{p\}$. Then, for $l\in {\mathbb {N}}$, there exists a vector $\nu '_l\in \text {cl}\,\eta _L(V'_l)$, such that $f_L(\nu '_l)\le f_M(\nu '_l)$. Since $\text {cl}\,\eta _L(V'_l)\searrow \eta _L(\{p\})=\{\nu \}$, it follows that $\nu '_l\rightarrow \nu$ and consequently,

$$\begin{aligned} f_L(\nu )\le f_M(\nu ). \end{aligned}$$

It remains to remove the extra assumption $V{\setminus } \{p\}\subseteq \text {int}\,M$. Let $L,M,V,p,\nu$ be as in the statement of Proposition 7.2. At this point we are going to assume that L is of class $C_+^2$ (the case where M is of class $C_+^2$ can be treated completely similarly and is left to the reader). Let $g:{\mathbb {S}}^{n-1}\rightarrow {\mathbb {R}}$ be a $C^2$ function which is strictly positive on ${\mathbb {S}}^{n-1}{\setminus }\{\nu \}$ and satisfies $g(\nu )=0$. Then, for small positive t, the function $h_L-tg$ is also a support function of class $C_+^2$. Set $L_t$ for the $C_+^2$ convex body whose support function equals $h_L-tg$. Then, $h_{L_t}\le h_L$, thus $L_t$ is contained in L. Furthermore, it holds $h_{L_t}(\nu )=h_L(\nu )$ and, therefore, $L_t$ is supported by the supporting hyperplane of L, whose outer unit normal vector is $\nu$. But since $L_t$ is contained in L, it follows that (for small t) $p\in \text {bd}\,L_t$. Moreover, $h_{L_t}(a)<h_L(a)$, for all $a\in W{\setminus }\{\nu \}=\eta _L(V){\setminus }\{\nu \}$, so $\eta ^{-1}_{L_t}(W){\setminus }\{p\}\subseteq \text {int}\,L\subseteq \text {int}\,M$. This, together with the fact that $p\in \text {bd}\,L_t\cap \text {bd}\,M$, shows that

$$\begin{aligned} f_{L_t}(\nu )\le f_M(\nu ), \end{aligned}$$

for small $t>0$. However, since L is of class $C^2_+$, (3) shows that $f_{L_t}(\nu )\xrightarrow {t\rightarrow 0^+}f_L(\nu )$ and the result follows. $\square$