Asymptotic theory of $C$-pseudo-cones¹¹1This paper was supported by the NSFC (No. 12371060), the Shaanxi Fundamental Science Research Project for Mathematics and Physics (No. 22JSZ012) and the Excellent Graduate Training Program of SNNU (No. LHRCCX23142).

The polar operator on the space $\mathscr{K}^{n}$ of convex bodies and its functional counterpart, the Legendre transform on the class $\text{Cvx}(\mathbb{R}^{n})$ of lower semi-continuous convex functions, play an important role in convex geometry, see [28] for more details. As an abstraction, an order reversing involution on certain partially ordered sets is called a duality. All dualities on $\mathscr{K}^{n}$ and $\text{Cvx}(\mathbb{R}^{n})$ are completely characterized by the polar operator [8] and the Legendre transform [4], respectively. In [3, 5], Artstein-Avidan and Milman introduced a new duality transform that is different from the Legendre transform on the sub-class $\text{Cvx}_{0}(\mathbb{R}^{n})$ (the class of geometric convex functions) of $\text{Cvx}(\mathbb{R}^{n})$. Naturally, a question arises: what is the geometric version corresponding to this new duality transform? In a special case, Rashkovskii [27] studied the copolarity of coconvex sets in the positive orthant of $\mathbb{R}^{n}$. Later, Artstein-Avidan, Sadovsky and Wyczesany [6] pointed out that all order reversing quasi-involutions can be induced by a cost function and exhibited various attractive dualities. Moreover, they systematically studied this geometric version of the new duality transform (termed dual polarity) and showed that the acting object of the dual polarity is a cone-like unbounded closed convex set. They also established the Blaschke-Santaló type inequality for the cone-like sets under the condition of essential symmetry. The cone-like set and dual polarity were further examined in [42], where the authors referred to this cone-like set as the pseudo-cone.

On the other hand, inspired by the works of Khovanskii and Timorin [19] and Milman and Rotem [25], Schneider [30] established the Brunn-Minkowski theory for $C$-close sets, where $C$ is a pointed closed convex cone in $\mathbb{R}^{n}$. Similar to the dual Minkowski problem posed by Huang, Lutwak, Yang and Zhang [16] for convex bodies, Li, Ye and the fourth author [22] introduced the concept of $C$-compatible sets and studied their copolarity to address the dual Minkowski problem for unbounded closed convex sets. Adopting terms and conventions from [33, 34], these related results have shown that dual polarity is equivalent to copolarity and that $C$-compatible sets are the same as $C$-pseudo-cones. Quite recently, Semenov and Zhao [37] studied some characterizations of the $C$-asymptotic sets to solve the Minkowski problem for the $C$-asymptotic sets given by the infinity measure. Inspired by these, our first result in this paper is a classification of $C$-pseudo-cones. That is, a non-degenerated $C$-pseudo-cone can be uniquely decomposed into the sum of a $C$-asymptotic set and a point in $C$. To state this result precisely, we gather some foundational concepts, which can be found in [6, 22, 30, 33, 34, 35, 42].

Denote by $\langle\cdot,\cdot\rangle$ and $|\cdot|$ the inner product and norm in $\mathbb{R}^{n}$, respectively. We write $B_{2}^{n}$ and $\mathbb{S}^{n-1}$ for the unit ball and the unit sphere in $\mathbb{R}^{n}$, respectively. The topological boundary and the topological interior of a subset $E\subset\mathbb{R}^{n}$ are denoted by $\partial E$ and $\text{int}\,E$, respectively. Denote by $H_{u}^{-}$ a negative half-space with the outer normal vector $u\in\mathbb{S}^{n-1}$. Let $C$ be a pointed closed convex cone in $\mathbb{R}^{n}$ with non-empty interior, i.e., $C$ is a $n$-dimensional closed convex cone that is line-free. The polar cone of $C$ is defined by $C^{\circ}=\{y\in\mathbb{R}^{n}\,|\,\langle x,y\rangle\leqslant 0,\,\forall\,x\in C\}$, which is also line-free. Denote by $\Omega_{C}=\mathbb{S}^{n-1}\cap\text{int}\,C$ and $\Omega_{C^{\circ}}=\mathbb{S}^{n-1}\cap\text{int}\,C^{\circ}$. Let $o\notin\mathbb{A}\subset C$ be an unbounded closed convex set and $A=C\setminus\mathbb{A}$. If $A$ has finite volume, then $\mathbb{A}$ is called a $C$-close set and $A$ is called a $C$-coconvex set. Specifically, a $C$-close set $\mathbb{A}$ is called a $C$-full set if $A$ is bounded. Moreover, $\mathbb{A}$ is called a $C$-asymptotic set if $\mathbb{A}$ is asymptotic to $\partial C$ at infinity, i.e.,

$$\lim_{x\in\partial\mathbb{A},|x|\rightarrow+\infty}d(x,\partial C)=0,$$

(1)

where $d(x,\partial C)$ is the distance from $x$ to $\partial C$. Let $E\subset\mathbb{R}^{n}$ be a non-empty closed convex set. The recession cone of $E$ is defined by ${\rm rec}\,E=\{x\in\mathbb{R}^{n}\,|\,E+x\subset E\}$. If $E$ is unbounded, then ${\rm rec}\,E$ is a closed convex cone. Let $o\notin E\subset\mathbb{R}^{n}$ be a non-empty closed convex set, $E$ is called a pseudo-cone if there holds $\lambda x\in E$ for any $\lambda\geqslant 1$ and $x\in\,E$. For an $n$-dimensional pointed closed convex cone $C$, if $E$ is a pseudo-cone and ${\rm rec}\,E=C$, then $E$ is called a $C$-pseudo-cone.

Now, we present a asymptotic characterization for the $C$-pseudo-cones. For the definition of the support function of $C$-pseudo-cones, please see (3).

We call a $C$-pseudo-cone satisfying $(i)$ or $(ii)$ in Theorem 1 as a non-degenerated $C$-pseudo-cone. Moreover, for a non-degenerated $C$-pseudo-cone $E=z+\mathbb{A}$, we call $z$ as the $C$-starting point of $E$ and $z+C$ as the asymptotic cone of $E$. If $z=o$, then Theorem 1 reduces to Theorem 2.6 in [37]. But the methods in this paper are different from [37]. There is a beautiful example of the degenerated $C$-pseudo-cone proposed by Schneider in high dimension ($n\geqslant 3$), please see Figure 1 in Section 3. Furthermore, from a new perspective, the $C$-starting point of a non-degenerated $C$-pseudo-cone can offer insights into the conic linear bundle. As applications of this asymptotic theory of $C$-pseudo-cones, one can establish a strengthened version of the Brunn-Minkowski inequality for $C$-coconvex sets [30], see Section 6 for details.

Since the main behavior of a $C$-pseudo-cone concentrates around the origin, i.e., all $C$-pseudo-cones have almost same asymptotic behavior at infinity, the usual Lebesgue measure space is not suitable for $C$-pseudo-cones. Following Schneider’s idea on the finiteness of the measure [34], one can endow a weight on $C$ to ensure the compactness for $C$-pseudo-cones. Throughout this paper, we denote the weight function by $\Theta:C\setminus\{o\}\rightarrow(0,\infty)$, which is a $(-q)$-homogeneous continuous function with $q\in\mathbb{R}$. A good example of such a weight function is $\Theta(y)=|y|^{-q}$ for any $y\in C\setminus\{o\}$.

The weighted surface area measure [34] of $E$ is given by the push-forward measure of the weighted boundary Hausdorff measure, specifically,

$$S^{\Theta}_{n-1}(E,\cdot)={\boldsymbol{\nu}_{E}}_{\sharp}\,\Theta(\mathscr{H}^{n-1}\llcorner\partial_{e}E),$$

where $\boldsymbol{\nu}_{E}$ is the Gauss image of $E$ and $\partial_{e}E=E\cap\text{cl}\,(C\setminus E)$ is the effective boundary of $E$ (see [22] or section 2 for more details). Schneider [34] showed that for $q>n-1$ and every $C$-pseudo-cone $E$, $S^{\Theta}_{n-1}(E,\cdot)$ is finite. Moreover, the weighted volume $V_{\Theta}(E)$ and the weighted co-volume $\overline{V}_{\Theta}(E)$ of $C$-pseudo-cone $E$ are defined as follows:

$$V_{\Theta}(E)=\int_{E}\Theta(x)\,d\mathscr{H}^{n}(x),\quad\overline{V}_{\Theta}(E)=\int_{C\setminus E}\Theta(x)\,d\mathscr{H}^{n}(x).$$

Schneider [34] also proved that for any $n-1<q<n$ and every $C$-pseudo-cone $E$, $\overline{V}_{\Theta}(E)$ is finite.

From the asymptotic perspective of a $C$-pseudo-cone, we define the asymptotic weighted co-volume functional of non-degenerated $C$-pseudo-cones as follows: Let $z$ be the $C$-starting point of the $C$-pseudo-cone $E$, expressed as $E=z+\mathbb{A}$ for some $C$-asymptotic set $\mathbb{A}$. The asymptotic weighted co-volume of $E$ is defined by

$$T_{\Theta}(E)=T_{\Theta}(\mathbb{A},z)=\int_{(z+C)\setminus E}\Theta(x)\,d\mathscr{H}^{n}(x).$$

We can view this asymptotic weighted co-volume $T_{\Theta}(E)$ as a functional of two variables. If we fix a $C$-asymptotic set $\mathbb{A}$, then $T_{\Theta}(E)=T_{\Theta}(\mathbb{A},\cdot)$ becomes a generalized function on $C$. As our second result, we summarize and enhance Schneider’s finiteness theory from [34] into the following results.

In the table above, the case of $n-1<q<n$ has been solved by Schneider [34]. Here, we focus more on the case of $q>n$. The highlighted (in blue) entries indicate that there is useful and interesting information regarding the finiteness of the asymptotic weighted co-volume and the weighted co-volume of the $C$-pseudo-cone $E$. This suggests that as $q$ exceeds $n$, we can gain additional insights into the structure and properties of $C$-pseudo-cones. For example, we demonstrate that for every $q>n$ and any $C$-asymptotic set $\mathbb{A}$, there exists a basic yet important convolution formula (see Section 4 for details):

$$T_{\Theta}(\mathbb{A},z)=\boldsymbol{\chi}_{-C}\ast\Theta\,(z)-V_{\Theta}(z+\mathbb{A}),\,z\in C,$$

where $\boldsymbol{\chi}_{-C}$ is the characteristic function of $-C$. Fixing a $C$-asymptotic set $\mathbb{A}$, $T_{\Theta}(\mathbb{A},\cdot)$ is a generalized function on $C$. We establish a decay estimate for $T_{\Theta}(\mathbb{A},\cdot)$ at infinity, and then we obtain the following interesting formula: Let $q>n$ and $o\neq z\in C$. If the weight function $\Theta$ is $C^{1}$-smooth, then

$$\boldsymbol{\chi}_{-C}\ast\Theta\,(z)=\frac{1}{n-q}\int_{z+C}\frac{\partial\Theta}{\partial z}(x)\,d\mathscr{H}^{n}(x).$$

We believe that this imperceptible formula will be instrumental in establishing the Brunn-Minkowski inequality for $T_{\Theta}$.

Based on the fact that the Minkowski sum of two $C$-pseudo-cones is also a $C$-pseudo-cone (see Lemma 3.9 in [37]), we can derive the following Brunn-Minkowski type inequality. For $0\leqslant q\leqslant n-1$, we call a $C$-pseudo-cone $E$ as a $(C,\Theta)$-close set if $\overline{V}_{\Theta}(E)<+\infty$.

From the asymptotic perspective of $C$-pseudo-cones, we conjecture that the following Brunn-Minkowski type inequality holds. Open problem (Asymptotic Brunn-Minkowski inequality for $C$-pseudo-cones): Let $E_{1},E_{2}$ be two non-degenerated $C$-pseudo-cones, and let $z_{1},z_{2}\neq o$ be their respective $C$-starting points. For $\lambda\in(0,1)$ and $0\leqslant q\neq n$, there holds

$$T_{\Theta}(\lambda E_{1}+(1-\lambda)E_{2})^{\frac{1}{n-q}}\leqslant\lambda T_{\Theta}(E_{1})^{\frac{1}{n-q}}+(1-\lambda)T_{\Theta}(E_{2})^{\frac{1}{n-q}}?$$

Related results of the noncompact version of the classical Minkowski problem have long been studied by Pogorelov [26], Bakelman [7], and Urbas [38]. Subsequently, Chou and Wang [12] studied the smooth solutions to noncompact Minkowski problems in detail. Recently, Choi et al. [9, 10, 11] studied significant problems related to the evolution of hypersurfaces asymptotic to a cylinder. In cases where these hypersurfaces are asymptotic to a cone, Schneider [30, 34] was the first to investigate the noncompact Minkowski problem. Moreover, concerning the weighted Minkowski problem for $C$-pseudo-cones, Schneider [34] provided an elegant result that characterizes the weighted surface area measure for $n-1<q<n$. As applications of our asymptotic theory, we present solutions to the weighted Minkowski problem for $0\leqslant q<n-1$ as follows.

The organization of this paper is as follows. In Section 2, we discuss the push-forward measure and co-area formula related to $C$-pseudo-cones. Section 3 presents the asymptotic theory of $C$-pseudo-cones. Section 4 summarizes results on the finiteness of the weighted co-volume and the asymptotic weighted co-volume. In Section 5, we provide integral formulas for the (asymptotic) weighted co-volume and the weighted surface area measure. Theorem 3 and the asymptotic Brunn-Minkowski inequality for $C$-pseudo-cones are discussed in Section 6. Finally, we study the weighted Minkowski problem with subcritical exponent.

In this section, we introduce some tools related to $C$-pseudo-cones. Let $E\subset\mathbb{R}^{n}$ be a non-empty closed convex set. The recession cone of $E$ is defined as ${\rm rec}\,E=\{x\in\mathbb{R}^{n}\,|\,E+x\subset E\}.$ If $E$ is unbounded, then ${\rm rec}\,E$ is a closed convex cone.

Let $E$ be a $C$-pseudo-cone. The support function of $E$ is defined by

$$h_{E}(v)=\max\{\langle x,v\rangle\,|\,x\in E\},\,v\in\Omega_{C^{\circ}}.$$

(3)

Note that $-\infty<h_{E}(v)<0$ for any $v\in\Omega_{C^{\circ}}$, we denote $\overline{h}_{E}=-h_{E}$. The radial function of $E$ is defined by

$$\rho_{E}(u)=\min\{r>0\,|\,ru\in E\},\,u\in\Omega_{C}.$$

Fixed $\mathfrak{u}\in\Omega_{C}\cap(-\Omega_{C^{\circ}})$, then $\langle\mathfrak{u},x\rangle>0$ for any $x\in C\setminus\{o\}$ (see e.g., [33]). Define $C(t)=\{x\in C\,|\,\langle x,\mathfrak{u}\rangle=t\}$ and $C^{-}(t)=\{x\in C\,|\,\langle x,\mathfrak{u}\rangle\leqslant t\}$.

Li, Ye and the fourth author [22] introduced the following notion of $C$-compatible set.

Let $E$ be a $C$-pseudo-cone. Some geometric mappings and sets associated with $E$, such as the Gauss image map and the effective boundary of $E$, as discussed in [22], are summarized as follows:

$$\qquad\quad\text{transport mappings}\left\{\begin{aligned} \boldsymbol{\nu}_{E}:\mathscr{B}(\partial_{e}E)&\rightarrow\mathscr{B}(\Omega_{C^{\circ}}^{e}),\,\beta\mapsto\boldsymbol{\nu}_{E}(\beta),\\ r_{E}:\Omega_{C}^{e}&\rightarrow\partial_{e}E,\,u\mapsto\rho_{E}(u)u,\\ \boldsymbol{\alpha}_{E}:\mathscr{B}(\Omega_{C}^{e})&\rightarrow\mathscr{B}(\Omega_{C^{\circ}}^{e}),\,\omega\mapsto\boldsymbol{\nu}_{E}(r_{E}(\omega)),\\ \end{aligned}\right.$$

$$\text{transport inverse mappings}\left\{\begin{aligned} \boldsymbol{\nu}^{*}_{E}:\mathscr{B}(\Omega_{C^{\circ}}^{e})&\rightarrow\mathscr{B}(\partial_{e}E),\,\eta\mapsto\boldsymbol{\nu}^{*}_{E}(\eta),\\ r^{-1}_{E}:\partial_{e}E&\rightarrow\Omega_{C}^{e},\,x\mapsto\frac{x}{|x|},\\ \boldsymbol{\alpha}^{*}_{E}:\mathscr{B}(\Omega_{C^{\circ}}^{e})&\rightarrow\mathscr{B}(\Omega_{C}^{e}),\,\eta\mapsto r^{-1}_{E}(\boldsymbol{\nu}^{*}_{E}(\eta)).\\ \end{aligned}\right.$$

For more details, one can refer to [22, p. 2018-2019]. Let $\Theta$ be a positive continuous function on $C\setminus\{0\}$. We consider two measures with density $\Theta$: the $\Theta$-weighted spherical Lebesgue measure, denoted by $\Theta\mu_{\mathbb{S}^{n-1}}\llcorner\Omega_{C}^{e}$ and the $\Theta$-weighted boundary Hausdorff measure, denoted by $\Theta\mathscr{H}^{n-1}\llcorner\partial_{e}E$. The measure $\Theta\mathscr{H}^{n-1}\llcorner\partial_{e}E$ can be pushed forward to the $\Theta$-weighted surface area measure $S^{\Theta}_{n-1}(E,\cdot)$ on $\Omega_{C^{\circ}}^{e}$ via the Gauss image map $\boldsymbol{\nu}_{E}$. Specifically, for any Borel set $\eta\in\mathscr{B}(\Omega_{C^{\circ}}^{e})$,

$$S^{\Theta}_{n-1}(E,\eta)={\boldsymbol{\nu}_{E}}_{\sharp}(\Theta\mathscr{H}^{n-1}\llcorner\partial_{e}E)\,(\eta)=\int_{\boldsymbol{\nu}^{*}_{E}(\eta)\cap\partial_{e}E}\Theta(x)\,d\mathscr{H}^{n-1}(x).$$

(4)

It is not hard to verify that $S^{\Theta}_{n-1}(E,\cdot)$ is a Radon measure on $\Omega_{C^{\circ}}^{e}$. By the standard approximation theory in measure theory, (4) is equivalent to the following: for any bounded measurable function $f$ on $\Omega_{C^{\circ}}^{e}$,

$$\int_{\Omega_{C^{\circ}}^{e}}f(v)\,dS^{\Theta}_{n-1}(E,v)=\int_{\partial_{e}E}f(\nu_{E}(x))\Theta(x)\,d\mathscr{H}^{n-1}(x),$$

(5)

where $\nu_{E}$ is the Gauss map of $E$ defined $\mathscr{H}^{n-1}$-a.e. on $\partial_{e}E$.

To transform the boundary integrals and the spherical integrals, one can apply the co-area formula of Federer [13, Theorem 3.2.22] and the approximation Jacobian of the radial map. For further details, please refer to [18, p. 170], [22, p. 2022], and [34, p. 9]. To ensure rigor, we provide a simple yet important lemma:

Due to Lemma 2, $r_{E}$ is differentiable almost everywhere on $\Omega_{C}^{e}$ and $r^{-1}_{E}$ is differentiable almost everywhere on $\partial_{e}E$. Let $v\in\Omega_{C^{\circ}}$ be a regular normal vector of $E$, $x=\boldsymbol{\nu}_{E}^{*}(v)$ and $u=r_{E}^{-1}(x)$. By establishing the orthonormal frame $\{e_{1},\cdots,e_{n-1}\}$ of $\mathbb{S}^{n-1}$ at $v$, we obtain the orthogonal decomposition:

$$\left\{\begin{aligned} u&=\sum_{i=1}^{n-1}\langle u,e_{i}\rangle e_{i}+\langle u,v\rangle v\triangleq\sum_{i=1}^{n-1}s_{i}e_{i}+sv,\\ x&=\rho_{E}(u)u=\sum_{i=1}^{n-1}\langle x,e_{i}\rangle e_{i}+\langle x,v\rangle v\triangleq\sum_{i=1}^{n-1}h_{i}e_{i}+h_{E}(v)v,\end{aligned}\right.$$

where $h_{i}$ is the covariant partial derivative of the support function $h$ with respect to the frame $\{e_{1},\cdots,e_{n-1}\}$. Denote by Id the identity mapping on $T_{x}\partial E$, then the tangent mapping of $r_{E}^{-1}$ at $x$ is given by (noting that $r^{-1}_{E}$ is differentiable $\mathscr{H}^{n-1}$-a.e. on $\partial_{e}E$):

$$d\Big{(}\frac{x}{|x|}\Big{)}=\frac{1}{|x|}(\text{Id}-u\otimes(u\otimes\text{Id})).$$

Therefore, using the Gram matrix of $\Big{\{}d\Big{(}\frac{x}{|x|}\Big{)}(e_{1}),\cdots,d\Big{(}\frac{x}{|x|}\Big{)}(e_{n-1})\Big{\}}$, we have

$$\Big{|}d\Big{(}\frac{x}{|x|}\Big{)}(e_{1})\wedge\cdots\wedge d\Big{(}\frac{x}{|x|}\Big{)}(e_{n-1})\Big{|}=\frac{\overline{h}}{|x|^{n}},$$

thus the $(\mathscr{H}^{n-1},n-1)$-approximate Jacobian of $r_{E}^{-1}$ at $\mathscr{H}^{n-1}$-a.e. $x\in\partial_{e}E$ is

$$ap\,J_{n-1}r_{E}^{-1}(x)=\Big{|}d\Big{(}\frac{x}{|x|}\Big{)}(e_{1})\wedge\cdots\wedge d\Big{(}\frac{x}{|x|}\Big{)}(e_{n-1})\Big{|}=\frac{|h_{E}(v)|}{|x|^{n}}=\frac{|\langle v,u\rangle|}{\rho_{E}^{n-1}(u)}.$$

Since $r_{E}\circ r_{E}^{-1}=\text{Id}$, the $(\mathscr{H}^{n-1},n-1)$-approximate Jacobian of $r_{E}$ at $\mu_{\mathbb{S}^{n-1}}$-a.e. $u\in\Omega_{C}^{e}$ is

$$ap\,J_{n-1}r_{E}(u)=(ap\,J_{n-1}r_{E}^{-1}(x))^{-1}=\frac{\rho_{E}^{n-1}(u)}{|\langle v,u\rangle|}=\frac{\rho_{E}^{n}(u)}{|\langle x,v\rangle|}.$$

Applying the co-area formula in [13, Theorem 3.2.22], for any integrable functions $f$ on $\Omega_{C}^{e}$ and $g$ on $\partial_{e}E$, we have

$$\left\{\begin{aligned} \int_{\Omega_{C}^{e}}f(u)\frac{\rho_{E}^{n}(u)}{\overline{h}_{E}(\alpha_{E}(u))}\,d\mu_{\mathbb{S}^{n-1}}(u)=\int_{\partial_{e}E}f\Big{(}\frac{x}{|x|}\Big{)}\,d\mathscr{H}^{n-1}(x),\\ \int_{\partial_{e}E}g(x)\frac{\overline{h}_{E}(\nu_{E}(x))}{|x|^{n}}\,d\mathscr{H}^{n-1}(x)=\int_{\Omega_{C}^{e}}g(r_{E}(u))\,d\mu_{\mathbb{S}^{n-1}}(u).\\ \end{aligned}\right.$$

(6)

As an unbounded closed convex set in $C$, the $C$-pseudo-cone is a central research object in unbounded Brunn-Minkowski theory. In this section, we will discuss some asymptotic properties of $C$-pseudo-cones. These properties are useful for establishing the Brunn-Minkowski theory for more general unbounded closed convex sets.

Let $E$ be a $C$-pseudo-cone and $p_{E}(o)$ be the metric projection of the origin $o$ on $E$, $u\in\partial C\cap\mathbb{S}^{n-1}$. Denote by $l_{x,u}=\{x+\lambda u\,|\,\forall\ \lambda>0,\ x\in\mathbb{R}^{n}\ \text{and}\ u\in\mathbb{S}^{n-1}\}$ the ray starting at $x\in\mathbb{R}^{n}$ in the direction $u\in\mathbb{S}^{n-1}$. If $E\subset x_{0}+C$ for some $x_{0}\in C$, we define the following set

$$\partial E|_{u}=\{x\in\partial E\,|\,\exists\lambda,\mu\geqslant 0\ \text{such that}\,x=x_{0}+\lambda(p_{E}(o)-x_{0})+\mu u\},$$

then there exists a boundary partition

$$\partial E=\bigcup_{u\in\partial C\cap\mathbb{S}^{n-1}}\partial E|_{u}.$$

Moreover, we define the function $f_{u,x_{0}}$ as $f_{u,x_{0}}(|x|)=d(x,l_{x_{0},u}),\ x\in\partial E|_{u}$.