# An affine Orlicz Pólya-Szegö principle

Youjiang Lin 111Research of the author is supported by the funds of the Basic
and Advanced Research Project of CQ CSTC cstc2015jcyjA00009 and Scientific and
Technological Research Program of Chongqing Municipal Education Commission
(KJ1500628), Scientific research funds of Chongqing Technology and Business
University 2015-56-02.

Abstract. The author [38] established the affine Orlicz Pólya-Szegö principle
for log-concave functions and conjectured that the principle can be extended
to the general Orlicz Sobolev functions. In this paper, we confirm this
conjecture completely. An affine Orlicz Pólya-Szegö principle, which includes
all the previous affine Pólya-Szegö principles as special cases, is formulated
and proved. As a consequence, an Orlicz-Petty projection inequality for star
bodies is established.

AMS Subject Classification 2010 46E35, 46E30, 52A40

Keywords and Phrases. Pólya-Szegö principle; Orlicz-Sobolev space; Steiner
symmetrization; Orlicz projection bodies

##  1 Introduction

The classical Pólya-Szegö principle [54] states that, given a non-negative
function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$, the Dirichlet integral
$\int_{\mathbb{R}^{n}}|\nabla f|^{p}$ decreases under suitable rearrangements,
the two most common of which are the symmetric decreasing rearrangement about
a point and Steiner symmetrization about a hyperplane. Their corresponding
Pólya-Szegö inequalities are a powerful tool to approach a wide number of
variational problems of geometric and functional nature (see, e.g.,[2, 7, 9,
12, 14, 15, 19, 58, 59]).

Zhang [63] and Lutwak et al. [47] formulated and proved a remarkable affine
$L_{p}$ Pólya-Szegö principle for $1\leq p<n$, which significantly strengthens
the classical Pólya-Szegö principle. Later, Cianchi et al. [17] perfectly
completed the picture of affine Pólya-Szegö principle (for general $p\geq 1$).
In [32], Haberl, Schuster and Xiao obtained a beautiful asymmetric version of
the affine Pólya-Szegö principle and proved that it is stronger and directly
implies the symmetric form of Cianchi et al.

The affine $L_{p}$ Pólya-Szegö-type principle is closely related to the
$L_{p}$ Brunn-Minkowski theory of convex bodies (see e.g., [5, 23, 30, 31, 32,
35, 40, 41, 42, 43, 44, 45, 46, 48, 60] for additional references). The fact
that $L_{p}$ spaces have a natural generalization, known as Orlicz spaces,
motivated Lutwak, Yang and Zhang [49, 50] to initiate an extension of the
$L_{p}$ Brunn-Minkowski theory to an Orlicz-Brunn-Minkowski theory. The
definition of a corresponding addition came later, in the work of Gardner, Hug
and Weil [25]. They developed a very general and comprehensive framework for
Orlicz-Brunn-Minkowski theory. This theory has expanded rapidly (see e.g., [4,
26, 27, 33, 36, 61, 62, 64, 65, 66]).

Let $\mathcal{N}$ be the class of convex functions
$\phi:\mathbb{R}\rightarrow[0,\infty)$ such that $\phi(0)=0$ and such that
$\phi$ is either strictly decreasing on $(-\infty,0]$ or $\phi$ is strictly
increasing on $[0,\infty)$. Moreover, we will assume throughout that
$\Phi(t)=\max\\{\phi(t),\phi(-t)\\}$, $t\in[0,\infty)$, and
$\lim_{t\rightarrow+\infty}\Phi(t)/t=+\infty$. It is easily checked that
$\Phi(t)$ is a convex function and strictly increasing on $[0,\infty)$.

Let $G$ denote a bounded open subset of $\mathbb{R}^{n}$ and $W^{1,\Phi}(G)$
the first order Orlicz-Sobolev space on $G$ (see Section 3). Let
$W_{0}^{1,\Phi}(G)$ denote the subspace of $W^{1,\Phi}(G)$ of those functions
whose continuation by $0$ outside $G$ belongs to $W^{1,\Phi}(\mathbb{R}^{n})$.
For $v\in S^{n-1}$ and $f\in W_{0}^{1,\Phi}(G)$, we define

$\displaystyle\|v\|_{f,\phi}=\|\nabla_{v}f\|_{\phi}=\inf\left\\{\lambda>0:\;\frac{1}{|G|}\int_{G}\phi\left(\frac{\nabla_{v}f}{\lambda}\right)dx\leq
1\right\\},$ (1.1)

where $\nabla_{v}f$ is the directional derivative of $f$ in the direction $v$.
The definition immediately provides the extension of $\|\cdot\|_{f,\phi}$ from
$S^{n-1}$ to $\mathbb{R}^{n}$. Now $(\mathbb{R}^{n},\|\cdot\|_{f,\phi})$ is
the $n$-dimensional Banach space that we shall associate with $f$. And its
unit ball $B_{\phi}(f)=\\{x\in\mathbb{R}^{n}:\|x\|_{f,\phi}\leq 1\\}$ is a
convex body in $\mathbb{R}^{n}$. An important fact is that its volume
$|B_{\phi}(f)|$ is invariant under affine transformations of the form
$x\mapsto Ax+x_{0}$, with $x_{0}\in\mathbb{R}^{n}$ and $A\in SL(n)$. We call
the unit ball $B_{\phi}(f)$ the Orlicz-Sobolev affine ball of $f$. We call

$\displaystyle\mathcal{E}_{\phi}(f):=|B_{\phi}(f)|^{-\frac{1}{n}}=\left(\frac{1}{n}\int_{S^{n-1}}\|\nabla_{v}f\|_{\phi}^{-n}dv\right)^{-\frac{1}{n}}$
(1.2)

the Orlicz-Sobolev affine energy of $f$.

Talenti [56] (also see [6]) proved a Euclidean Orlicz Pólya-Szegö principle,

$\displaystyle\|\nabla f^{\star}\|_{\phi}\leq\|\nabla f\|_{\phi},$ (1.3)

which extended the classical Pólya-Szegö principle to Orlicz-Sobolev spaces.

In [38], the author established an affine Orlicz Pólya-Szegö principle for
log-concave function: If $f\in W_{0}^{1,\Phi}(G)$ is a log-concave functions
and $f^{\star}$ is its symmetric decreasing rearrangement, then

$\displaystyle\mathcal{E}_{\phi}(f^{\star})\leq\mathcal{E}_{\phi}(f),$ (1.4)

which includes all the affine $L_{p}$ Pólya-Szegö principles as special cases
(when restricted to log-concave functions). The author [38] conjectured that
the principle can be extended to the general Orlicz Sobolev functions. In this
paper, we confirm this conjecture completely.

######  Theorem 1.1.

If $f\in W_{0}^{1,\Phi}(G)$ and $f^{\star}$ is its symmetric decreasing
rearrangement, then

$\displaystyle\mathcal{E}_{\phi}(f^{\star})\leq\mathcal{E}_{\phi}(f).$ (1.5)

When $\phi(t)=(1-\lambda)(t)_{+}^{p}+\lambda(t)_{-}^{p}$, where $p>1$,
$\lambda\in[0,1]$, $(t)_{+}:=\max\\{t,0\\}$ and $(t)_{-}:=\max\\{-t,0\\}$, the
affine Orlicz Pólya-Szegö principle becomes the general affine Pólya-Szegö-
type principle (established in [53]). The symmetric affine Pólya-Szegö
principle [17] and the asymmetric affine Pólya-Szegö principle [32] correspond
to the cases of $\lambda=1/2$ and $\lambda=0$, respectively.

##  2 Outline of the proof

The proof of the affine $L_{p}$ Pólya-Szegö principle in [47] mainly relies on
the affine $L_{p}$ Petty projection inequality from [46] and the solution of
the normalized $L_{p}$ Minkowski problem [44]. Haberl, Schuster and Xiao in
[32] mainly relies on the Haberl-Schuster version of the affine isoperimetric
inequality [30] and the solution of the normalized $L_{p}$ Minkowski problem
[44] to prove the asymmetric affine Pólya-Szegö principle. The discovery of
the Haberl-Schuster version of the $L_{p}$ Petty projection inequality was
made possible through the important advances in valuation theory by Ludwig
[39, 40]. Inspired by the work of Haddad, Jiménez, and Montenegro [34], Nguyen
[53] made use of the general $L_{p}$ Busemann-Petty centroid inequality [46]
to prove the general affine Pólya-Szegö-type principle.

As mentioned in [38], since the function $\phi$ defining the Orlicz-Sobolev
spaces is usually not multiplicative, i.e., $\phi(xy)\neq\phi(x)\phi(y)$ for
$x,y\in\mathbb{R}$, and the Orlicz Minkowski problem has not been completely
solved, we can not use the Orlicz-Petty projection inequality [49] and the
solution of the even Orlicz Minkowski problem [29] to prove the affine Orlicz
Pólya-Szegö principle. In [38], making use of the functional Steiner
symmetrization, the author gave a direct proof of the Orlicz Pólya-Szegö
principle for log-concave functions. In this paper, we still make use of the
functional Steiner symmetrization to the affine Orlicz Pólya-Szegö principle.
Comparing with the paper [38], in this paper we mainly overcome the following
several difficulties.

Firstly, in [38], the convexity property of the level sets of a log-concave
function plays a role in proving the Orlicz Pólya-Szegö principle. But for the
general $W_{0}^{1,\Phi}(G)$ functions, the level sets of functions are sets
more general than convex bodies. Thus, it is crucial to figure out the
structure of the level sets of $W_{0}^{1,\Phi}(G)$. In this paper, we show
that the level sets of $W_{0}^{1,\Phi}(G)$ functions are sets of finite
perimeter and define the Steiner symmetrization of the level sets in a
different way (see Section 4.3), which makes it possible to make use of the
functional Steiner symmetrization to prove the affine Orlicz Pólya-Szegö
principle.

Secondly, since log-concave functions are co-area regular, by [7, Theorem 2],
the symmetric decreasing rearrangements of log-concave functions can be
approximated in the strong $W^{1,p}$-topology by a sequence of Steiner
symmetrizations. Therefore, the Orlicz-Sobolev affine ball operator
$B_{\phi}:W_{0}^{1,\Phi}(G)\rightarrow\mathcal{K}_{o}^{n}$ is continuous for
log-concave functions. But for the general Sobolev functions, the Orlicz-
Sobolev affine ball operator may be discontinuous for the co-area irregular
functions. In this paper, making use of the weak convergence of Steiner
symmetrizations of $f\in W^{1,\phi}(\mathbb{R}^{n})$, we prove that the
Orlicz-Sobolev affine ball operator is weakly continuous in such a way that
there exist a convex body $K_{0}$ and a sequence of successive Steiner
symmetrizations $\\{f_{k}\\}_{k\geq 0}$ of $f$ such that $B_{\phi}(f_{i})$
converges to $K_{0}$ and $K_{0}\subset B_{\phi}(f^{\star})$ (see Lemma 5.5).

Thirdly, in [38, Lemma 5.2], for log-concave functions, the author proved that
$\nabla f$ and $\nabla Sf$ are equal to the same constant in the line
parallelling to $e_{n}$ on the boundaries of subgraphs of $f$ and $Sf$. For
general Sobolev functions, without the convexity, we can not make use of Lemma
1.5.14 and Theorem 1.5.15 in [55] with respect to convex functions to prove
the similar conclusion. In this paper, using the more subtle method, we prove
that Sobolev functions have the similar properties as log-concave functions
(see Lemma 6.2).

Functional Steiner symmetrization has turned out to be very fruitful in
proving isoperimetric theorems in analysis and function theory (see. e.g., [3,
7, 8, 10, 13, 11, 15, 16, 22, 37, 38, 57], and the references therein). In the
beautiful paper [15], Cianchi and Fucso analyzed the case of equality in
Steiner symmetrization inequalities for Dirichlet-type integrals. Specially,
Cianchi and Fucso [15] proved the following Pólya-Szegö principle for $f\in
W_{0}^{1,1}(G)$:

$\int_{SG}\psi(\nabla Sf)dx\leq\int_{G}\psi(\nabla f)dx,$

where $Sf$ is the Steiner symmetrization of $f$ and $\psi$ is a convex
function from $\mathbb{R}^{n}$ into $[0,+\infty)$, vanishing at $0$. Comparing
with Cianchi-Fucso’s method—using the Steiner symmetrization of one-
dimensional restrictions of Sobolev functions and Fubini’s theorem, we mainly
make use of the Steiner symmetrization of the level sets and the co-area
formula.

##  3 Notation and preliminary results

The setting will be Euclidean $n$-space $\mathbb{R}^{n}$ with origin $o$. We
write $e_{1},\dots,e_{n-1},e_{n}$ for the standard orthonormal basis of
$\mathbb{R}^{n}$ and when we write
$\mathbb{R}^{n}=\mathbb{R}^{n-1}\times\mathbb{R}$ we always assume that
$e_{n}$ is associated with the last factor.

We will attempt to use $x,y$ for vectors in $\mathbb{R}^{n}$ and $x^{\prime}$,
$y^{\prime}$ for vectors in $\mathbb{R}^{n-1}$, and $u,v\in S^{n-1}$ for unit
vectors in $\mathbb{R}^{n}$. Let $u^{\perp}$ denote the $n$-dimensional linear
subspace orthogonal to $u$ in $\mathbb{R}^{n}$. Let $l_{u}$ denote the line
that passes through the origin and is parallel to $u$. We will use
$a,b,s,t,\alpha$ for numbers in $\mathbb{R}$ and $c,\lambda$ for strictly
positive reals. If $Q$ is a Borel subset of $\mathbb{R}^{n}$ and $Q$ is
contained in an $i$-dimensional affine subspace of $\mathbb{R}^{n}$ but in no
affine subspace of lower dimension, then $|Q|$ will denote the $i$-dimensional
Lebesgue measure of $Q$. Let $\mathcal{H}^{n}$ denote the $n$-dimensional
Hausdorff measures. Let $B_{n}$ denote the Euclidean unit ball in
$\mathbb{R}^{n}$. Let $\omega_{n}$ denote the $n$-dimensional volume of the
unit ball in $\mathbb{R}^{n}$. If $x\in\mathbb{R}^{n}$ then by abuse of
notation we will write $|x|=\sqrt{x\cdot x}$.

For $A\in SL(n)$ write $A^{t}$ for the transpose of $A$ and $A^{-t}$ for the
inverse of the transpose of $A$. We write $|A|$ for the absolute value of the
determinant of $A$.

###  3.1 Convex bodies and star bodies

In this section we fix our notation and collect basic facts from convex
geometry. General references for the theory of convex bodies are the books by
Gardner [24], Gruber [28], Schneider [55]. We write $\mathcal{K}^{n}$ for the
set of convex bodies (compact convex subsets) of $\mathbb{R}^{n}$. We write
$\mathcal{K}^{n}_{o}$ for the set of convex bodies that contain the origin in
their interiors. A compact set $K\subset\mathbb{R}^{n}$ is a star-shaped set
(with respect to the origin) if the intersection of every straight line
through the origin with $K$ is a line segment. Let $K\subset\mathbb{R}^{n}$ be
a compact star shaped set (with respect to the origin), the radial function
$\rho(K,\cdot):\mathbb{R}^{n}\backslash\\{0\\}\rightarrow\mathbb{R}$ is
defined by

$\displaystyle\rho(K,x)=\rho_{K}(x)=\max\\{\lambda\geq 0:\;\lambda x\in K\\}.$
(3.1)

If $\rho_{K}$ is strictly positive and continuous, then we call $K$ a star
body (with respect to the origin). In what follows we will denote the class of
star bodies (with respect to the origin) in $\mathbb{R}^{n}$ by
$\mathcal{S}_{o}^{n}$.

For $K\in\mathcal{K}^{n}$, let
$h(K;\cdot)=h_{K}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ denote the support
function of $K$; i.e.,

$h(K;x):=\max\\{x\cdot y:\;y\in K\\}.$

Thus, if $y\in\partial K$, then

$\displaystyle h_{K}(v_{K}(y))=v_{K}(y)\cdot y,$ (3.2)

where $v_{K}(y)$ denotes an outer unit normal to $K$ at $y\in\partial K$.

For $K,L\in\mathcal{K}^{n}$, the Hausdorff distance of $K$ and $L$ is defined
by

$\displaystyle\delta(K,L):=\sup_{u\in S^{n-1}}|h_{K}(u)-h_{L}(u)|.$ (3.3)

If $K\in\mathcal{K}_{o}^{n}$, then the polar body $K^{\ast}$ is defined by

$K^{\ast}:=\\{x\in\mathbb{R}^{n}:x\cdot y\leq 1\;{\rm for\;all}\;y\in K\\}.$

For $K\in\mathcal{K}_{o}^{n}$, it is easily verified that

$\displaystyle h_{K^{\ast}}=1/\rho_{K}\;\;\;{\rm
and}\;\;\;\rho_{K^{\ast}}=1/h_{K}.$ (3.4)

For $K\in\mathcal{S}_{o}^{n}$, the function
$g_{K}(\cdot)=\|\cdot\|_{K}:\mathbb{R}^{n}\rightarrow[0,\infty)$ defined by

$\displaystyle g_{K}(x)=\|x\|_{K}=\inf\\{\lambda>0:\;x\in\lambda K\\}$ (3.5)

is called the gauge function of $K$.

By (3.1) and (3.5), it is clear that for $K\in\mathcal{S}_{o}^{n}$,

$\displaystyle g_{K}=1/\rho_{K}.$ (3.6)

By (3.5), for $x\in\mathbb{R}^{n}$ and $K\in\mathcal{S}_{o}^{n}$, it follows
immediately that

$\displaystyle g_{K}(x)=1\;\;{\rm if\;and\;only\;if}\;\;x\in\partial K.$ (3.7)

Let $K\in\mathcal{S}_{o}^{n}$ be a star body whose radial function is positive
and locally Lipschitz continuous. Since $g_{K}=\frac{1}{\rho_{K}}$ and there
exist $0<c_{1}<c_{2}$ such that $c_{1}<\rho_{K}(u)<c_{2}$ for any $u\in
S^{n-1}$, $g_{K}$ is positive and locally Lipschitz continuous on
$\mathbb{R}^{n}$. Therefore, for almost all $y\in\partial K$, the gradient
$\nabla g_{K}(y)$ exists. For $y\in\partial K$, $g_{K}(y)=1$. Differentiating
this equation gives $\nabla g_{K}(y)\cdot dy=0$. Thus, $\nabla g_{K}(y)$ is
orthogonal to the tangent hyperplane of $\partial K$ at $y$. For $r>0$,
differentiating the equation $g_{K}(ry)=rg_{K}(y)$ with respect to $r$ gives

$\displaystyle y\cdot\nabla g_{K}(y)=g_{K}(y).$ (3.8)

Therefore, $|\nabla g_{K}(y)|\neq 0$. The unit outer normal vector $v_{K}(y)$
to $K$ at $y\in\partial K$ is given by

$\displaystyle v_{K}(y)=\frac{\nabla g_{K}(y)}{|\nabla g_{K}(y)|}.$ (3.9)

Moreover, for $y\in\partial K$, by (3.9), (3.8) and $g_{K}(y)=1$, we have

$\displaystyle y\cdot v_{K}(y)=y\cdot\frac{\nabla g_{K}(y)}{|\nabla
g_{K}(y)|}=\frac{1}{|\nabla g_{K}(y)|}.$ (3.10)

By (3.9) and (3.10), for $y\in\partial K$, we have

$\displaystyle\nabla g_{K}(y)=\frac{v_{K}(y)}{y\cdot v_{K}(y)}.$ (3.11)

Moreover, since that $g_{K}(ry)=rg_{K}(y)$ for $r>0$, setting $x=ry$, we
obtain

$\displaystyle\nabla g_{K}(x)=\nabla g_{K}(y).$ (3.12)

###  3.2 $BV(\mathbb{R}^{n})$ functions and sets of finite perimeter

In this section, we review some basic definitions and facts about functions of
bounded variation on $\mathbb{R}^{n}$. Good general references for this are
Ambrosio, Fusco and Pallara [1], Evans and Gariepy[20], Ziemer[67].

Let $C_{c}^{1}(\mathbb{R}^{n})$ stand for the compactly supported continuously
differentiable functions on $\mathbb{R}^{n}$. Let $f\in
L^{1}(\mathbb{R}^{n})$; we say that $f$ is a function of bounded variation of
$\mathbb{R}^{n}$ if the weak derivative of $f$ is representable by a finite
Radon measure on $\mathbb{R}^{n}$, i.e. if

$\int_{\mathbb{R}^{n}}f\frac{\partial\phi}{\partial
x_{i}}dx=-\int_{\mathbb{R}^{n}}\phi dD_{i}f\;\;{\rm for\;all\;}\phi\in
C_{c}^{1}(\mathbb{R}^{n}),\;i=1,\dots,n$

for some $\mathbb{R}^{n}$-valued measure $Df=(D_{1}f,\dots,D_{n}f)$ on
$\mathbb{R}^{n}$. The vector space of all functions of bounded variation in
$\mathbb{R}^{n}$ is denoted by $BV(\mathbb{R}^{n})$.

A Borel set $E\subset\mathbb{R}^{n}$ is said to have finite perimeter in an
open set $\Omega$ provided that the characteristic function of $E$,
$\mathcal{X}_{E}$, is a function of bounded variation in $\Omega$. Thus, the
partial derivatives of $\mathcal{X}_{E}$ are Radon measures in $\Omega$ and
the perimeter of $E$ in $\Omega$, $P(E,\Omega)$, is defined as

$P(E,\Omega)=\|D\mathcal{X}_{E}\|(\Omega).$

A set $E$ is said to be of locally finite perimeter if $P(E,\Omega)<\infty$
for every bounded open set $\Omega$. If $E$ is of finite perimeter in
$\mathbb{R}^{n}$, it is simply called a set of finite perimeter. Usually, we
write $P(E)$ instead of $P(E,\mathbb{R}^{n})$.

Let $\mathcal{C}^{n}$ denote the set of sets of locally finite perimeter. By
${\rm cl}E$, ${\rm int}E$ and $\partial E$ we denote, respectively, the
closure, interior and boundary of $E\in\mathcal{C}^{n}$. Let $E^{\prime}_{u}$
denote the image of the orthogonal projection of $E$ onto $u^{\perp}$. The
essential projection of a set $E\subset\mathcal{C}^{n}$ onto $u^{\perp}$ is
defined as

$E_{u}^{\prime,+}=\\{x^{\prime}\in u^{\perp}:\;|(l_{u}+x^{\prime})\cap
E|>0\\}.$

For $E,F\in\mathcal{C}^{n}$, the symmetric difference of $E$ and $F$ is

$E\triangle F=(E\backslash F)\cup(F\backslash E).$

Let $B(x,r)$ denote the ball, centered at $x$, having radius $r$. Let $E$ be
of locally finite perimeter. The reduced boundary of $E$, $\partial^{\ast}E$,
consists of all points $x\in\mathbb{R}^{n}$ for which the following hold:

(i) $\|D\mathcal{X}_{E}\|[B(x,r)]>0$ for all $r>0$,

(ii) If $v_{E,r}(x):=-D\mathcal{X}_{E}[B(x,r)]/\|D\mathcal{X}_{E}\|[B(x,r)]$,
then the limit $v_{E}(x)=\lim_{r\rightarrow 0}v_{E,r}(x)$ exists with
$|v_{E}(x)|=1$.

$v_{E}(x)$ is called the generalized exterior normal to $E$ at
$x\in\partial^{\ast}E$. By [1, Theroem 3.59], if $E\in\mathcal{C}^{n}$,
$\partial^{\ast}E$ is $n-1$-rectifiable, i.e., it can be covered, except for
an $\mathcal{H}^{n-1}$-negligible subset, by countably many
$(n-1)$-dimensional surfaces of class $C^{1}$.

If $E\subset\mathbb{R}^{n}$ is a Lebesgue measurable set, the measure-
theoretic boundary of $E$ is defined by

$\partial^{M}E=\left\\{x:\limsup_{r\rightarrow 0}\frac{|B(x,r)\cap
E|}{r^{n}}>0\;{\rm and}\;\limsup_{r\rightarrow
0}\frac{|B(x,r)-E|}{r^{n}}>0\right\\}.$

By Federer’s structure theorem (see, e.g., [20, Lemma 1 on Page 208] and [1,
Theroem 3.61]), if $E\in\mathcal{C}^{n}$, then
$\partial^{\ast}E\subset\partial^{M}E$ and
$\mathcal{H}^{n-1}(\partial^{M}E-\partial^{\ast}E)=0$.

For $E\in\mathcal{C}^{n}$, let

$D_{E}:=\sup\\{|x-y|:\;x,y\in E\\}$

denote the diameter of $E$. Since $E$ is obviously contained in the right
cylinder whose base is $E^{\prime}_{u}$ and whose height is $D_{E}$, then we
have the crude estimate

$\displaystyle\int_{\partial^{\ast}E}(u\cdot
v_{E}(y))_{+}d\mathcal{H}^{n-1}(y)\geq\frac{|E|}{D_{E}}.$ (3.13)

###  3.3 Orlicz-Sobolev spaces

In this section we summarize the necessary definitions and results about
Orlicz-Sobolev spaces. For a detailed account of these facts, the reader could
consult the books of Maz’ya [51, 52] and the paper of Cianchi [18].

Let $\phi\in\mathcal{N}$ and $\Phi(t)=\max\\{\phi(t),\phi(-t)\\}$,
$t\in[0,\infty)$. Let $G$ be an open subset of $\mathbb{R}^{n}$. The Orlicz
space $L^{\Phi}(G)$ is defined as

$\displaystyle L^{\Phi}(G)$ $\displaystyle=$ $\displaystyle\left\\{f:\;f\;{\rm
is\;a\;Lebesgue\;measurable\;real\;valued\;function\;on}\;G\right.$ (3.14)
$\displaystyle\left.{\rm
such\;that}\;\int_{G}\Phi\left(\frac{|f(x)|}{\lambda}\right)dx<\infty\;{\rm
for\;some\;}\lambda>0\right\\}.$

The Luxemburg norm $\|f\|_{L^{\Phi}(G)}$ is defined as

$\displaystyle\|f\|_{L^{\Phi}(G)}=\inf\left\\{\lambda>0:\int_{G}\Phi\left(\frac{|f(x)|}{\lambda}\right)dx\leq
1\right\\}.$ (3.15)

The space $L^{\Phi}(G)$, equipped with the norm $\|\cdot\|_{L^{\Phi}}$, is a
Banach space. Note that, if $\Phi(s)=s^{p}$ and $p>1$, then
$L^{\Phi}(G)=L^{p}(G)$, the usual $L_{p}$ space, and
$\|\cdot\|_{L^{\Phi}(G)}=\|\cdot\|_{L^{p}(G)}$. Usually, we write
$\|\cdot\|_{\Phi}$ instead of $\|\cdot\|_{L^{\Phi}}$.

The first order Orlicz-Sobolev space $W^{1,\Phi}(G)$ is defined as

$\displaystyle W^{1,\Phi}(G)=\\{f\in L^{\Phi}(G):f\;{\rm is}\;{\rm
weakly}\;{\rm differentiable}\;{\rm and\;}|\nabla f|\in L^{\Phi}(G)\\}.$
(3.16)

Here, $\nabla$ denotes the weak gradient. The space $W^{1,\Phi}(G)$, equipped
with the norm

$\displaystyle\|f\|_{W^{1,\Phi}(G)}=\|f\|_{\Phi}+\||\nabla f|\|_{\Phi},$
(3.17)

is a Banach space. Clearly, $W^{1,\Phi}(G)=W^{1,p}(G)$, the standard Sobolev
space, if $\Phi(s)=s^{p}$ with $p>1$.

$W_{0}^{1,\Phi}(G)$ will denote the subspace of $W^{1,\Phi}(G)$ of those
functions whose extension by $0$ outside $G$ belongs to
$W^{1,\Phi}(\mathbb{R}^{n})$. If $f\in W_{0}^{1,\Phi}(G)$, then $f$ vanishes
on the boundary of $G$ and we let

$\bar{f}(x)=\left\\{\begin{aligned} &f(x),&x\in G&\\\
&0,&x\in\mathbb{R}^{n}/G,&\end{aligned}\right.$ (3.18)

then $\bar{f}\in W^{1,\Phi}(\mathbb{R}^{n})$. Throughout this paper, $\bar{f}$
will denote the extension of $f$ by $0$ outside $G$.

For the nonnegative function $f\in W_{0}^{1,\Phi}(G)$, we define the subgraph
of $f$ by

$\displaystyle{\rm sub}f:=\\{(x,x_{n+1})\in\mathbb{R}^{n+1}:x\in
G,\;0<x_{n+1}<f(x)\\}.$ (3.19)

And we define its superlevel sets by

$\displaystyle[f]_{h}:=\\{x\in G:\;f(x)\geq h\\},\;h>0.$ (3.20)

######  Lemma 3.1.

If $f\in W_{0}^{1,\Phi}(G)$, then for $\mathcal{L}^{1}$-a.e. $h>0$, $[f]_{h}$
is a set of finite perimeter.

###### Proof.

By [38, Lemma 3.3], if $f\in W_{0}^{1,\Phi}(G)$, then $\bar{f}\in
W^{1,1}(\mathbb{R}^{n})$. Since Sobolev space $W^{1,1}(\mathbb{R}^{n})$ is the
subspace of $BV(\mathbb{R}^{n})$ (see e.g. [1, Page 118]), $\bar{f}\in
BV(\mathbb{R}^{n})$. By [1, Theorem 3.40], if $f\in BV(\mathbb{R}^{n})$, then
the superlevel set $[f]_{h}$ is of finite perimeter for $\mathcal{L}^{1}$-a.e.
$h>0$. ∎

We shall make use of the fact that for $\phi\in\mathcal{N}$,
$a_{i}\in\mathbb{R}$ and $b_{i}>0$, $i=1,2,\cdots,m$,

$\displaystyle\sum_{i=1}^{m}b_{i}\cdot\phi\left(\frac{\sum_{i=1}^{m}a_{i}}{\sum_{i=1}^{m}b_{i}}\right)\leq\sum_{i=1}^{m}\left(b_{i}\phi\left(\frac{a_{i}}{b_{i}}\right)\right).$
(3.21)

This is a trivial consequence of the convexity of $\phi$.

We shall make use of the following clear fact.

######  Lemma 3.2.

If $\phi\in\mathcal{N}$, for $a,b\in\mathbb{R}$ and $a\neq 0$, then the
function

$\displaystyle\Psi(t):=\phi(at-b)+\phi(-at-b),\;t>0$ (3.22)

is increasing.

##  4 Steiner symmetrization

###  4.1 Steiner symmetrization of compact sets

Without loss of generality, we only need to consider the Steiner
symmetrization in the direction $e_{n}$.

Given any compact subset $E$ of $\mathbb{R}^{n}$, define, for
$x^{\prime}\in\mathbb{R}^{n-1}$,

$\displaystyle E_{x^{\prime}}:=\\{x_{n}\in\mathbb{R}:(x^{\prime},x_{n})\in
E\\}.$ (4.1)

Hereafter, $\mathcal{L}^{m}$ denote the outer Lebesgue measure in
$\mathbb{R}^{m}$ and $E^{\prime}$ denote the image of the orthogonal
projection of $E$ onto $e_{n}^{\perp}$. Then, we define the Steiner symmetral
$SE$ of $E$ about the hyperplane $e_{n}^{\perp}$ as

$SE:=\\{(x^{\prime},x_{n})\in\mathbb{R}^{n}:x^{\prime}\in
E^{\prime},\;|x_{n}|\leq\mathcal{L}^{1}(E_{x^{\prime}})/2\\}.$

Moreover, Steiner symmetrization preserves volume, i.e., $|SE|=|E|$.

When considering the convex body $K\in\mathcal{K}_{o}^{n}$ as
$K\subset\mathbb{R}^{n-1}\times\mathbb{R}$, the Steiner symmetral, $SK$, of
$K$ in the direction $e_{n}$ is given by

$\displaystyle
SK=\left\\{\left(x^{\prime},\frac{1}{2}t+\frac{1}{2}s\right)\in\mathbb{R}^{n-1}\times\mathbb{R}:(x^{\prime},t),(x^{\prime},-s)\in
K\right\\},$ (4.2)

and its boundary is given by

$\displaystyle\partial
SK=\left\\{\left(x^{\prime},\frac{1}{2}t+\frac{1}{2}s\right)\in\mathbb{R}^{n-1}\times\mathbb{R}:(x^{\prime},t),(x^{\prime},-s)\in\partial
K\;{\rm with}\;t\neq-s\right\\}.$ (4.3)

In this paper, we shall make critical use of the following fact that follows
directly from (4.2), (4.3), and (3.7).

######  Lemma 4.1.

Suppose $K,L\in\mathcal{K}_{o}^{n}$ and consider
$K,L\subset\mathbb{R}^{n-1}\times\mathbb{R}$. Then

$SK\subset L,$

if and only if

$\|(x^{\prime},t)\|_{K}=1=\|(x^{\prime},-s)\|_{K},\;\;with\;t\neq-s\;\;\Longrightarrow\;\;\|(x^{\prime},t/2+s/2)\|_{L}\leq
1.$

###  4.2 Steiner symmetrization of functions

Let $\mathcal{X}_{A}(x)$ denote the characteristic function of
$A\subset\mathbb{R}^{n}$, i.e.,

$\mathcal{X}_{A}(x):=\left\\{\begin{aligned} &1&&{\rm if}\;x\in A,\\\ &0&&{\rm
if}\;x\in\mathbb{R}^{n}\backslash A.\end{aligned}\right.$ (4.4)

######  Definition 4.1.

For nonnegative function $f\in W_{0}^{1,\Phi}(G)$, the Steiner symmetrization,
$Sf$, of $f$ with respect to $e_{n}$ is defined as

$\displaystyle Sf(x)=\int_{0}^{\infty}\mathcal{X}_{S[f]_{h}}(x)dh,\;\;x\in
SG.$ (4.5)

By Definition 4.1, it is easily checked that

$\displaystyle[Sf]_{h}\;{\rm is\;equivalent\;to}\;S[f]_{h},\;\;{\rm
for}\;h>0.$ (4.6)

Thus, we have for the subgraphs

$\displaystyle{\rm sub}(Sf)\;{\rm is\;equivalent\;to}\;S({\rm sub}f).$ (4.7)

The following lemma was given in [15, Theorem 2.1].

######  Lemma 4.2.

If $f\in W_{0}^{1,\Phi}(G)$, then $Sf\in W_{0}^{1,\Phi}(SG)$.

###  4.3 The steiner symmetrization of the level sets of $W_{0}^{1,\Phi}(G)$
functions

For $f\in W_{0}^{1,\Phi}(G)$ and any $h>0$, $[f]_{h}$ is a set of finite
perimeter. Let $[f]^{\prime}_{h}$ denote the image of orthogonal projection of
$[f]_{h}$ onto $e^{\perp}_{n}$. For a fixed $x^{\prime}\in[f]_{h}^{\prime}$,
let $[f]_{h}(x^{\prime})$ denote the set
$\\{x_{n}\in\mathbb{R}:(x^{\prime},x_{n})\in[f]_{h}\\}$. Let

$\displaystyle
N([f]_{h},x^{\prime}):=\frac{1}{2}\mathcal{H}^{0}\left(\partial^{\ast}([f]_{h}(x^{\prime}))\right),$
(4.8)

where $\mathcal{H}^{0}$ is the counting measure. Since $[f]_{h}$ is a set of
finite perimeter, by [15, Theorem D], for $\mathcal{L}^{n-1}$-a.e.
$x^{\prime}\in[f]^{\prime}_{h}$,
$\mathcal{H}^{0}\left(\partial^{\ast}([f]_{h}(x^{\prime}))\right)$ is a finite
even number.

For fixed positive integer $i$, let

$\displaystyle[f]^{\prime}_{h,i}:=\\{x^{\prime}\in[f]_{h}^{\prime}:\;N([f]_{h},x^{\prime})=i\\}.$
(4.9)

Let

$\displaystyle[f]_{h,i}:=\\{(x^{\prime},x_{n})\in[f]_{h}:\;x^{\prime}\in[f]_{h,i}^{\prime}\\}.$
(4.10)

For the fixed $i$ and $k=1,2,\dots,i$, let
$\underline{\ell}^{k}_{h,i}:[f]_{h,i}^{\prime}\rightarrow\mathbb{R}$ and
$\overline{\ell}^{k}_{h,i}:[f]_{h,i}^{\prime}\rightarrow\mathbb{R}$ denote the
$k$-th undergraph function and $k$-th overgraph function of $[f]_{h,i}$ with
respect to $e_{n}$; i.e.,

$\displaystyle[f]_{h,i}=\bigcup_{k=1}^{i}\left\\{(x^{\prime},x_{n})\in\mathbb{R}^{n}:\;x^{\prime}\in[f]_{h,i}^{\prime},\;-\underline{\ell}^{k}_{h,i}(x^{\prime})\leq
x_{n}\leq\overline{\ell}^{k}_{h,i}(x^{\prime})\right\\}$ (4.11)

up to an $\mathcal{L}^{n}$-negligible set.

For the Steiner symmetral, $S[f]_{h,i}$, of $[f]_{h,i}$ in direction $e_{n}$,
we see that the image of the orthogonal projections onto $e_{n}^{\perp}$ of
both $[f]_{h,i}$ and $S[f]_{h,i}$ are identical. Let
$\overline{\varrho}_{h,i}$ and $\underline{\varrho}_{h,i}$ denote the
overgraph and undergraph functions of $S[f]_{h,i}$ with respect to $e_{n}$,
i.e.,

$\displaystyle
S[f]_{h,i}=\left\\{(x^{\prime},x_{n})\in\mathbb{R}^{n}:\;x^{\prime}\in[f]_{h,i}^{\prime},\;-\underline{\varrho}_{h,i}(x^{\prime})\leq
x_{n}\leq\overline{\varrho}_{h,i}(x^{\prime})\right\\}.$ (4.12)

Then, by the definition of the Steiner symmetrization of compact sets, we have

$\displaystyle\underline{\varrho}_{h,i}(x^{\prime})=\overline{\varrho}_{h,i}(x^{\prime})=\frac{1}{2}\sum_{k=1}^{i}\left(\underline{\ell}^{k}_{h,i}(x^{\prime})+\overline{\ell}^{k}_{h,i}(x^{\prime})\right),\;\;\;x^{\prime}\in[f]^{\prime}_{h,i}.$
(4.13)

For $f\in W_{0}^{1,\Phi}(G)$ and a Borel function $g:G\rightarrow\mathbb{R}$,
the following co-area formula (see, e.g., [51, Theorem 1.2.4] or [15, (3.7)])
is an important tool in proving our main theorem.

$\displaystyle\int_{\\{x\in G:\nabla f(x)\neq
0\\}}g(x)dx=\int_{0}^{\infty}\int_{\partial^{\ast}[f]_{h}}g(x)|\nabla
f(x)|^{-1}d\mathcal{H}^{n-1}(x)dh.$ (4.14)

Since $[f]_{h}$ is a set of finite perimeter, $\partial^{\ast}[f]_{h}$
consists of an $\mathcal{H}^{n-1}$-negligible subset and countably many
$(n-1)$-dimensional surfaces of class $C^{1}$.

Let

$\bar{\partial}^{\ast}[f]_{h}=\left\\{x\in\partial^{\ast}[f]_{h}:v_{[f]_{h}}(x)\cdot
e_{n}=0\right\\}$

and

$\breve{\partial}^{\ast}[f]_{h}=\left\\{x\in\partial^{\ast}[f]_{h}:v_{[f]_{h}}(x)\cdot
e_{n}\neq 0\right\\}.$

We will make use of the easily established fact that for a continuous function
$g:\partial^{\ast}[f]_{h}\rightarrow\mathbb{R}$,

$\displaystyle\int_{\breve{\partial}^{\ast}[f]_{h}}g(x)d\mathcal{H}^{n-1}(x)$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{\infty}\sum_{k=1}^{i}\int_{{\rm
int}[f]_{h,i}^{\prime}}g(x^{\prime},\overline{\ell}^{k}_{h,i}(x^{\prime}))\sqrt{1+|\nabla\overline{\ell}^{k}_{h,i}(x^{\prime})|^{2}}dx^{\prime}$
$\displaystyle+\sum_{i=1}^{\infty}\sum_{k=1}^{i}\int_{{\rm
int}[f]_{h,i}^{\prime}}g(x^{\prime},-\underline{\ell}^{k}_{h,i}(x^{\prime}))\sqrt{1+|\nabla\underline{\ell}^{k}_{h,i}(x^{\prime})|^{2}}dx^{\prime}.$

###  4.4 Approximation of the symmetric decreasing rearrangement by Steiner
symmetrizations

Let $G$ be a bounded open set in $\mathbb{R}^{n}$. Its symmetric rearrangement
$G^{\star}$ is the open centered ball whose volume agrees with that of $G$,

$G^{\star}:=\\{x\in\mathbb{R}^{n}:\;\omega_{n}|x|^{n}<|G|\\}.$

For $f\in W_{0}^{1,\Phi}(G)$, we define the symmetric decreasing rearrangement
$f^{\star}$ of $f$ by symmetrizing its level sets, that is

$\displaystyle
f^{\star}(x)=\int_{0}^{\infty}\mathcal{X}_{[f]^{\star}_{h}}(x)dh,\;\;\;x\in
G^{\star}.$ (4.16)

For Sobolev spaces $W^{1,p}(\mathbb{R}^{n})$, Burchard [7, Proposition 7.1]
proved the following proposition on the approximation of spherical
symmetrization by Steiner symmetrizations.

######  Proposition 4.1.

(Convergence of the $W^{1,1}$-norm). Let $f$ be a nonnegative function in
$W^{1,p}(\mathbb{R}^{n})$, $n\geq 2$ and $p\geq 1$, that vanishes at infinity.
There exists a sequence of successive Steiner symmetrizations
$\\{f_{k}\\}_{k\geq 0}$ of $f$ so that

$f_{k}\rightharpoonup f^{\star}\;{\rm weakly\;in}\;W^{1,1}.$

######  Remark 4.1.

For Orlicz-Sobolev spaces $W^{1,\Phi}(\mathbb{R}^{n})$, there does not exist a
result similar to that of Proposition 4.1. Thus we consider the problem in
$W^{1,p}(\mathbb{R}^{n})$. Since $f\in W_{0}^{1,1}(G)$ for $f\in
W_{0}^{1,\Phi}(G)$, there exists a sequence of successive Steiner
symmetrizations $\\{\bar{f}_{k}\\}_{k\geq 0}$ of $\bar{f}$ so that

$\bar{f}_{k}\rightharpoonup\bar{f}^{\star}\;{\rm
weakly\;in}\;W^{1,1}(\mathbb{R}^{n}).$

##  5 Definition and basic properties of Orlicz-Sobolev affine balls

The Orlicz-Sobolev affine ball $B_{\phi}(f)$ of $f\in W_{0}^{1,\Phi}(G)$ is
defined as the unit ball of the $n$-dimensional Banach space whose norm is
given by

$\displaystyle\|y\|_{f,\phi}=\inf\left\\{\lambda>0:\;\frac{1}{|G|}\int_{G}\phi\left(\frac{y\cdot\nabla
f(x)}{\lambda}\right)dx\leq 1\right\\},\;y\in\mathbb{R}^{n}.$ (5.1)

And the volume of the Orlicz-Sobolev affine ball is given by

$\displaystyle|B_{\phi}(f)|=\frac{1}{n}\int_{S^{n-1}}\|v\|_{f,\phi}^{-n}dv,$
(5.2)

where $dv$ denotes the spherical Lebesgue measure.

Since $f\in W_{0}^{1,\Phi}(G)$, it is impossible that there exists some
$u_{0}\in S^{n-1}$ such that $\nabla f(x)\cdot u_{0}\geq 0$ for almost all
$x\in G$. Since $\phi$ is strictly increasing on $[0,\infty)$ or strictly
decreasing on $(-\infty,0]$ it follows that for $y\neq 0$ the function

$\lambda\mapsto\frac{1}{|G|}\int_{G}\phi\left(\frac{y\cdot\nabla
f(x)}{\lambda}\right)dx$

is strictly decreasing in $(0,\infty)$. Thus, we have the following lemma.

######  Lemma 5.1.

If $f\in W_{0}^{1,\Phi}(G)$ and $y_{0}\in\mathbb{R}^{n}\backslash\\{0\\}$,
then

$\frac{1}{|G|}\int_{G}\phi\left(\frac{y_{0}\cdot\nabla
f(x)}{\lambda_{0}}\right)dx=1$

if and only if

$\|y_{0}\|_{f,\phi}=\lambda_{0}.$

The following Lemma 5.2, Lemma 5.3 and Lemma 5.4 demonstrate the affine
invariance of $\mathcal{E}_{\phi}(f)$, non-negativity and boundedness of
$\|\cdot\|_{f,\phi}$, respectively.

######  Lemma 5.2.

([38, Lemma 4.2]). If $f\in W_{0}^{1,\Phi}(G)$, then $\mathcal{E}_{\phi}(f)$
is invariant under $SL(n)$ transformations.

######  Lemma 5.3.

([38, Lemma 4.3]). If $f\in W_{0}^{1,\Phi}(G)$, then $\|\cdot\|_{f,\phi}$
defines a norm on the Banach space $(\mathbb{R}^{n},\|\cdot\|_{f,\phi})$. In
particular, $\|v\|_{f,\phi}>0$ for any $v\in S^{n-1}$.

######  Lemma 5.4.

([38, Lemma 4.4]). If $f\in W_{0}^{1,\Phi}(G)$ and let

$\displaystyle c_{\phi}=\max\left\\{c>0:\max\\{\phi(c),\phi(-c)\\}\leq
1\right\\},$ (5.3)

then for any $v\in S^{n-1}$, we have

$\displaystyle\frac{\int_{G}f(x)dx}{c_{\phi}|G|D_{G}}\leq\|v\|_{f,\phi}\leq\frac{\sup\\{|\nabla
f(x)|:x\in G\\}}{c_{\phi}}.$ (5.4)

The following lemma shows that the Orlicz-Sobolev affine ball operator
$B_{\phi}:W_{0}^{1,\Phi}(G)\rightarrow\mathcal{K}_{o}^{n}$ is weakly
continuous in some sense.

######  Lemma 5.5.

For $f_{i}\in W_{0}^{1,\Phi}(G_{i})$, $i=0,1,2,\dots$, if

$\displaystyle\bar{f}_{i}\rightharpoonup\bar{f}_{0},{\rm
weakly\;in}\;W^{1,1}(\mathbb{R}^{n}),$ (5.5)

then there exists a subsequence of $\\{B_{\phi}(f_{i})\\}_{i=1}^{\infty}$,
denoted by $\\{B_{\phi}(f_{i})\\}_{i=1}^{\infty}$ as well, and a convex body
$K_{0}$ such that $o\in K_{0}$,

$\displaystyle\lim_{i\rightarrow\infty}\delta(B_{\phi}(f_{i}),K_{0})=0$ (5.6)

and

$\displaystyle K_{0}\subset B_{\phi}(f_{0}).$ (5.7)

###### Proof.

For $u_{0}\in S^{n-1}$, let

$\displaystyle\|u_{0}\|_{f_{i},\phi}=\lambda_{i},$ (5.8)

and note that Lemma 5.4 gives

$\displaystyle
0<\frac{\int_{G_{i}}f_{i}(x)dx}{c_{\phi}|G_{i}|D_{G_{i}}}\leq\lambda_{i}.$
(5.9)

Moreover, by (5.5), we have

$\displaystyle\lim_{i\rightarrow\infty}\frac{\int_{G_{i}}f_{i}(x)dx}{c_{\phi}|G_{i}|D_{G_{i}}}=\frac{\int_{G_{0}}f_{0}(x)dx}{c_{\phi}|G_{0}|D_{G_{0}}}>0,$
(5.10)

which implies that there exists a real number $m>0$ such that
$\|u\|_{f_{i},\phi}>m$ for any $u\in S^{n-1}$ and any positive integer $i$.
Thus the radial functions $\rho(B_{\phi}(f_{i}),u)<\frac{1}{m}$ for any $u\in
S^{n-1}$ and any positive integer $i$. By Blaschke selection theorem (see [55,
Theorem 1.8.7]), there exists a subsequence of
$\\{B_{\phi}(f_{i})\\}_{i=1}^{\infty}$, denoted by
$\\{B_{\phi}(f_{i})\\}_{i=1}^{\infty}$ as well, that converges to a convex
body $K_{0}\in\mathcal{K}^{n}$. By Lemma 5.3, $\|u\|_{f_{i},\phi}>0$ for any
$u\in S^{n-1}$ and any positive integer $i$. Thus $o\in K_{0}$.

Let $\|u_{0}\|_{K_{0}}=\lambda_{\ast}$. By (5.6) and (5.8), we have

$\displaystyle\lim_{i\rightarrow\infty}\lambda_{i}=\lambda_{\ast}.$ (5.11)

Let $\tilde{f_{i}}=\bar{f_{i}}/\lambda_{i}$. Since
$\lambda_{i}\rightarrow\lambda_{\ast}$ and
$\bar{f}_{i}\rightharpoonup\bar{f}_{0}$ weakly in $W^{1,1}(\mathbb{R}^{n})$,
we have

$\displaystyle\tilde{f}_{i}\rightharpoonup\bar{f}_{0}/\lambda_{\ast}\;\;{\rm
weakly\;in}\;\;W^{1,1}(\mathbb{R}^{n}).$ (5.12)

The fact that $\|u_{0}\|_{f_{i},\phi}=\lambda_{i}$, together with Lemma 5.1
and (3.18), shows that

$\displaystyle\frac{1}{|G_{i}|}\int_{\mathbb{R}^{n}}\phi\left(u_{0}\cdot\nabla\tilde{f}_{i}(x)\right)dx=1\;{\rm
for\;all}\;i.$ (5.13)

Since $\phi$ is a convex function, by [21, Theorem 1 in P.19], the convex
gradient integral

$\frac{1}{|G|}\int_{\mathbb{R}^{n}}\phi\left(u_{0}\cdot\nabla\bar{f}(x)\right)dx$

is lower semicontinuous with respect to weak convergence in
$W^{1,1}(\mathbb{R}^{n})$. By (5.12) and (5.13), we have

$\frac{1}{|G_{0}|}\int_{\mathbb{R}^{n}}\phi\left(\frac{u_{0}\cdot\nabla\bar{f}_{0}(x)}{\lambda_{\ast}}\right)dx\leq\lim_{i\rightarrow\infty}\frac{1}{|G_{i}|}\int_{\mathbb{R}^{n}}\phi\left(u_{0}\cdot\nabla\tilde{f}_{i}(x)\right)dx=1.$

This, together with (3.18) and the definition (5.1) yields

$\displaystyle\|u_{0}\|_{f_{0},\phi}\leq\lambda_{\ast}=\|u_{0}\|_{K_{0}}.$
(5.14)

By (5.14) and the arbitrariness of $u_{0}\in S^{n-1}$, we have $K_{0}\subset
B_{\phi}(f_{0})$. ∎

##  6 Proof of the main theorem

######  Lemma 6.1.

If $f\in W_{0}^{1,\Phi}(G)$ and $x^{\prime}\in{\rm int}[f]^{\prime}_{h,i}$,
then

$\displaystyle\frac{\left|\left(-\nabla\overline{\varrho}_{h,i}(x^{\prime}),1\right)\right|}{\left|\nabla
Sf\left(x^{\prime},\overline{\varrho}_{h,i}(x^{\prime})\right)\right|}=\frac{1}{2}\sum_{k=1}^{i}\frac{\left|(-\nabla\underline{\ell}^{k}_{h,i}(x^{\prime}),-1)\right|}{\left|\nabla
f\left(x^{\prime},-\underline{\ell}^{k}_{h,i}(x^{\prime})\right)\right|}+\frac{1}{2}\sum_{k=1}^{i}\frac{\left|\left(-\nabla\overline{\ell}_{h,i}^{k}(x^{\prime}),1\right)\right|}{\left|\nabla
f\left(x^{\prime},\overline{\ell}^{k}_{h,i}(x^{\prime})\right)\right|}.$ (6.1)

###### Proof.

In fact, the vectors
$\left(-\nabla\overline{\varrho}_{h,i}(x^{\prime}),1\right)$ and $\nabla
Sf\left(x^{\prime},\overline{\varrho}_{h,i}(x^{\prime})\right)$ have the same
direction, i.e., the direction of the exterior normal vector of $[Sf]_{h}$ at
the point
$(x^{\prime},\overline{\varrho}_{h,i}(x^{\prime}))\in\partial^{\ast}[Sf]_{h}$.
Thus for the left side of (6.1) we have

$\displaystyle\frac{\left|\left(-\nabla\overline{\varrho}_{h,i}(x^{\prime}),1\right)\right|}{\left|\nabla
Sf\left(x^{\prime},\overline{\varrho}_{h,i}(x^{\prime})\right)\right|}=\frac{1}{\left|\frac{\partial
Sf}{\partial x_{n}}(x^{\prime},\overline{\varrho}_{h,i}(x^{\prime}))\right|}.$
(6.2)

Similarly, the vectors
$\left(-\nabla\overline{\ell}_{h,i}^{k}(x^{\prime}),1\right)$ and $\nabla
f\left(x^{\prime},\overline{\ell}^{k}_{h,i}(x^{\prime})\right)$ have the same
direction, i.e., the direction of the exterior normal vector to $[f]_{h}$ at
the point
$(x^{\prime},\overline{\ell}^{k}_{h,i}(x^{\prime}))\in\partial^{\ast}[f]_{h}$,
we have

$\displaystyle\frac{\left|\left(-\nabla\overline{\ell}_{h,i}^{k}(x^{\prime}),1\right)\right|}{\left|\nabla
f\left(x^{\prime},\overline{\ell}^{k}_{h,i}(x^{\prime})\right)\right|}=\frac{1}{\left|\frac{\partial
f}{\partial
x_{n}}\left(x^{\prime},\overline{\ell}^{k}_{h,i}(x^{\prime})\right)\right|}.$
(6.3)

Moreover, we have

$\displaystyle\frac{\left|(-\nabla\underline{\ell}^{k}_{h,i}(x^{\prime}),-1)\right|}{\left|\nabla
f\left(x^{\prime},-\underline{\ell}^{k}_{h,i}(x^{\prime})\right)\right|}=\frac{1}{\left|\frac{\partial
f}{\partial
x_{n}}\left(x^{\prime},-\underline{\ell}^{k}_{h,i}(x^{\prime})\right)\right|}.$
(6.4)

By (6.2), (6.3) and (6.4), (6.1) is equivalent to

$\displaystyle\frac{1}{\left|\frac{\partial Sf}{\partial
x_{n}}(x^{\prime},\overline{\varrho}_{h,i}(x^{\prime}))\right|}=\frac{1}{2}\sum_{k=1}^{i}\frac{1}{\left|\frac{\partial
f}{\partial
x_{n}}\left(x^{\prime},-\underline{\ell}^{k}_{h,i}(x^{\prime})\right)\right|}+\frac{1}{2}\sum_{k=1}^{i}\frac{1}{\left|\frac{\partial
f}{\partial
x_{n}}\left(x^{\prime},\overline{\ell}^{k}_{h,i}(x^{\prime})\right)\right|}.$
(6.5)

By [15, Lemma 4.1] (also see [7, (5.1)]), (6.5) is established. ∎

Since $Sf$ is symmetric with respect to $e_{n}^{\perp}$, i.e.,
$Sf(x^{\prime}+re_{n})=Sf(x^{\prime}-re_{n})$ for any $x^{\prime}+re_{n}\in
SG$, it is easy to check that for $x^{\prime}\in{\rm int}[f]^{\prime}_{h,i}$,

$\displaystyle\frac{\left|\left(-\nabla\overline{\varrho}_{h,i}(x^{\prime}),1\right)\right|}{\left|\nabla
Sf\left(x^{\prime},\overline{\varrho}_{h,i}(x^{\prime})\right)\right|}$
$\displaystyle=$
$\displaystyle\frac{\left|\left(-\nabla\underline{\varrho}_{h,i}(x^{\prime}),-1\right)\right|}{\left|\nabla
Sf\left(x^{\prime},-\underline{\varrho}_{h,i}(x^{\prime})\right)\right|}.$
(6.6)

######  Lemma 6.2.

If $f\in W_{0}^{1,\Phi}(G)$ and let

$G_{1}=\left\\{x\in G:\;\frac{\partial f}{\partial x_{n}}(x)=0\right\\}$

and

$G_{2}=\left\\{x\in SG:\;\frac{\partial Sf}{\partial x_{n}}(x)=0\right\\},$

then for any $z\in\mathbb{R}^{n}$, we have

$\displaystyle\int_{G_{1}}\phi(z\cdot\nabla
f(x))dx=\int_{G_{2}}\phi(z\cdot\nabla Sf(x))dx.$ (6.7)

###### Proof.

By [15, Proposition 2.3], if $f\in W_{0}^{1,1}(G)$, then for
$\mathcal{L}^{n-1}$-a.e. $x^{\prime}\in G^{\prime}$,

$\displaystyle|\\{x_{n}:(x^{\prime},x_{n})\in G,\;\frac{\partial f}{\partial
x_{n}}(x^{\prime},x_{n})=0\\}|=|\\{x_{n}:(x^{\prime},x_{n})\in
SG,\;\frac{\partial Sf}{\partial x_{n}}(x^{\prime},x_{n})=0\\}|.$ (6.8)

By (6.8), if $\mathcal{L}^{n}(G_{1})=0$, then $\mathcal{L}^{n}(G_{2})=0$. Thus
(6.7) is established.

If $\mathcal{L}^{n}(G_{1})>0$, then by (6.8)
$\mathcal{L}^{n}(G_{2})=\mathcal{L}^{n}(G_{1})>0$. Moreover, by (6.8), the
essential projections of $G_{1}$ and $G_{2}$ onto $e_{n}^{\perp}$ satisfy
$G_{1}^{\prime,+}=G_{2}^{\prime,+}$. Next, we prove that for
$\mathcal{L}^{n-1}$-a.e. $x^{\prime}\in G_{1}^{\prime,+}$,

$\displaystyle\int_{\\{x_{n}:\;(x^{\prime},x_{n})\in
G_{1}\\}}\phi\left(z\cdot\nabla
f(x^{\prime},x_{n})\right)dx_{n}=\int_{\\{x_{n}:\;(x^{\prime},x_{n})\in
G_{2}\\}}\phi\left(z\cdot\nabla Sf(x^{\prime},x_{n})\right)dx_{n}.$ (6.9)

For fixed $x^{\prime}\in G_{1}^{\prime,+}$, let

$\displaystyle D_{1}=\\{h\in\mathbb{R}:\;h=f(x^{\prime},x_{n})\;{\rm
and}\;(x^{\prime},x_{n})\in G_{1}\\}$ (6.10)

and

$\displaystyle D_{2}=\\{h\in\mathbb{R}:\;h=Sf(x^{\prime},x_{n})\;{\rm
and}\;(x^{\prime},x_{n})\in G_{2}\\}.$ (6.11)

Let $f^{x^{\prime}}(x_{n}):=f(x^{\prime},x_{n})$ and
$(Sf)^{x^{\prime}}(x_{n}):=Sf(x^{\prime},x_{n})$. Since ${\rm
sub}((Sf)^{x^{\prime}})$ is equivalent to $S({\rm sub}f^{x^{\prime}})$, we
have $D_{1}=D_{2}$. For $h\in D_{1}$, let

$D^{h}_{x^{\prime}}=\\{x_{n}:\;(x^{\prime},x_{n})\in
G_{1},\;f(x^{\prime},x_{n})=h\\}$

and

$\tilde{D}^{h}_{x^{\prime}}=\\{x_{n}:\;(x^{\prime},x_{n})\in
G_{2},\;Sf(x^{\prime},x_{n})=h\\}.$

Next, we prove that for any $h\in D_{1}$,

$\displaystyle\int_{D^{h}_{x^{\prime}}}\phi\left(z\cdot\nabla
f(x^{\prime},x_{n})\right)dx_{n}=\int_{\tilde{D}^{h}_{x^{\prime}}}\phi\left(z\cdot\nabla
Sf(x^{\prime},x_{n})\right)dx_{n}.$ (6.12)

By [7, Lemma 4.3], for $\mathcal{L}^{1}$-a.e. $x_{n}\in D_{x^{\prime}}^{h}$,
$\nabla f(x^{\prime},x_{n})$ is equal to a constant. By [7, Lemma 5.1], for
$\mathcal{L}^{1}$-a.e. $x_{n}\in\tilde{D}_{x^{\prime}}^{h}$, $\nabla
Sf(x^{\prime},x_{n})$ equals the same constant as $\nabla f(x^{\prime},x_{n})$
for $x_{n}\in D_{x^{\prime}}^{h}$. Thus by
$|D^{h}_{x^{\prime}}|=|\tilde{D}^{h}_{x^{\prime}}|$, (6.12) is established. By
(6.12) and the arbitrariness of $h\in D_{1}$, (6.9) is established. By (6.9)
and Fubini’s theorem, (6.7) is established. ∎

######  Proposition 6.1.

If $f\in W_{0}^{1,\Phi}(G)$, then

$\displaystyle S(B_{\phi}(f))\subset B_{\phi}(Sf).$ (6.13)

###### Proof.

Let

$\|(y^{\prime},t)\|_{f,\phi}=1\;\;{\rm
and}\;\;\|(y^{\prime},-s)\|_{f,\phi}=1,$

with $t\neq-s$. By Lemma 5.1, this means that

$\displaystyle\frac{1}{|G|}\int_{G}\phi\left((y^{\prime},t)\cdot\nabla
f(x)\right)dx=1$ (6.14)

and

$\displaystyle\frac{1}{|G|}\int_{G}\phi\left((y^{\prime},-s)\cdot\nabla
f(x)\right)dx=1.$ (6.15)

By Lemma 4.1, the desired inclusion will been established if we can show that

$\displaystyle\|(y^{\prime},t/2+s/2)\|_{Sf,\phi}\leq 1.$ (6.16)

For $x^{\prime}\in{\rm int}[f]^{\prime}_{h,i}$ and $k=1,2,\cdots,i$, let

$\displaystyle
c^{k}_{h,i}=\frac{|(-\nabla\overline{\ell}^{k}_{h,i}(x^{\prime}),1)|}{|\nabla
f(x^{\prime},\overline{\ell}^{k}_{h,i}(x^{\prime}))|}\;\;{\rm
and}\;\;d^{k}_{h,i}=\frac{|(-\nabla\underline{\ell}^{k}_{h,i}(x^{\prime}),-1)|}{|\nabla
f(x^{\prime},-\underline{\ell}^{k}_{h,i}(x^{\prime}))|}.$ (6.17)

For $x^{\prime}\in{\rm int}[f]^{\prime}_{h,i}$, let

$\displaystyle
a_{1}=\frac{|(-\nabla\overline{\varrho}_{h,i}(x^{\prime}),1)|}{|\nabla
Sf(x^{\prime},\overline{\varrho}_{h,i}(x^{\prime}))|}\;\;{\rm
and}\;\;a_{2}=\frac{|(-\nabla\underline{\varrho}_{h,i}(x^{\prime}),-1)|}{|\nabla
Sf(x^{\prime},-\underline{\varrho}_{h,i}(x^{\prime}))|}.$ (6.18)

By (6.6) and Lemma 6.1,

$\displaystyle
a_{1}=a_{2}=\frac{1}{2}\sum_{k=1}^{i}c^{k}_{h,i}+\frac{1}{2}\sum_{k=1}^{i}d^{k}_{h,i}.$
(6.19)

Since $\nabla Sf(x^{\prime},\overline{\varrho}_{h,i}(x^{\prime}))$ and
$(-\nabla\overline{\varrho}_{h,i}(x^{\prime}),1)$ have the same directions,

$\displaystyle\nabla
Sf(x^{\prime},\overline{\varrho}_{h,i}(x^{\prime}))=\frac{(-\nabla\overline{\varrho}_{h,i}(x^{\prime}),1)}{a_{1}}.$
(6.20)

Similarly, we have

$\displaystyle\nabla
Sf(x^{\prime},-\underline{\varrho}_{h,i}(x^{\prime}))=\frac{(-\nabla\underline{\varrho}_{h,i}(x^{\prime}),-1)}{a_{2}},$
(6.21) $\displaystyle\nabla
f(x^{\prime},\overline{\ell}^{k}_{h,i}(x^{\prime}))=\frac{(-\nabla\overline{\ell}^{k}_{h,i}(x^{\prime}),1)}{c^{k}_{h,i}},\;\;k=1,2,\dots,i$
(6.22)

and

$\displaystyle\nabla
f(x^{\prime},-\underline{\ell}^{k}_{h,i}(x^{\prime}))=\frac{(-\nabla\underline{\ell}^{k}_{h,i}(x^{\prime}),-1)}{d^{k}_{h,i}},\;\;k=1,2,\dots,i.$
(6.23)

By (4.3), (6.20), (6.21), (4.13), Lemma 3.2 and (6.19), we have that

$\displaystyle\int_{\partial^{\ast}[Sf]_{h}/G_{2}}\phi\left((y^{\prime},t/2+s/2)\cdot\nabla
Sf(x)\right)|\nabla Sf(x)|^{-1}d\mathcal{H}^{n-1}(x)$ $\displaystyle=$
$\displaystyle\int_{([f]_{h}/G_{1})^{\prime}}\phi\left((y^{\prime},t/2+s/2)\cdot\nabla
Sf(x^{\prime},\overline{\varrho}_{h}(x^{\prime}))\right)a_{1}dx^{\prime}$
$\displaystyle+\int_{([f]_{h}/G_{1})^{\prime}}\phi\left((y^{\prime},t/2+s/2)\cdot\nabla
Sf(x^{\prime},-\underline{\varrho}_{h}(x^{\prime}))\right)a_{2}dx^{\prime}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{\infty}\int_{{\rm
int}[f]^{\prime}_{h,i}}\phi\left(\frac{1}{2a_{1}}\left(t+s-y^{\prime}\cdot\sum_{k=1}^{i}\nabla(\overline{\ell}^{k}_{h,i}+\underline{\ell}^{k}_{h,i})(x^{\prime})\right)\right)a_{1}dx^{\prime}$
$\displaystyle+\sum_{i=1}^{\infty}\int_{{\rm
int}[f]^{\prime}_{h,i}}\phi\left(\frac{1}{2a_{2}}\left(-t-s-y^{\prime}\cdot\sum_{k=1}^{i}\nabla(\overline{\ell}^{k}_{h,i}+\underline{\ell}^{k}_{h,i})(x^{\prime})\right)\right)a_{2}dx^{\prime}$
$\displaystyle\leq$ $\displaystyle\sum_{i=1}^{\infty}\int_{{\rm
int}[f]^{\prime}_{h,i}}\phi\left(\frac{1}{2a_{1}}\left(i(t+s)-y^{\prime}\cdot\sum_{k=1}^{i}\nabla(\overline{\ell}^{k}_{h,i}+\underline{\ell}^{k}_{h,i})(x^{\prime})\right)\right)a_{1}dx^{\prime}$
$\displaystyle+\sum_{i=1}^{\infty}\int_{{\rm
int}[f]^{\prime}_{h,i}}\phi\left(\frac{1}{2a_{2}}\left(i(-t-s)-y^{\prime}\cdot\sum_{k=1}^{i}\nabla(\overline{\ell}^{k}_{h,i}+\underline{\ell}^{k}_{h,i})(x^{\prime})\right)\right)a_{2}dx^{\prime}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{i=1}^{\infty}\int_{{\rm
int}[f]^{\prime}_{h,i}}\phi\left(\frac{\sum_{k=1}^{i}(t-y^{\prime}\cdot\nabla\overline{\ell}^{k}_{h,i}(x^{\prime}))+\sum_{k=1}^{i}(s-y^{\prime}\cdot\nabla\underline{\ell}^{k}_{h,i}(x^{\prime}))}{\sum_{k=1}^{i}(c^{k}_{h,i}+d^{k}_{h,i})}\right)\sum_{k=1}^{i}(c^{k}_{h,i}+d^{k}_{h,i})dx^{\prime}$
$\displaystyle+\frac{1}{2}\sum_{i=1}^{\infty}\int_{{\rm
int}[f]^{\prime}_{h,i}}\phi\left(\frac{\sum_{k=1}^{i}(-t-y^{\prime}\cdot\nabla\underline{\ell}^{k}_{h,i}(x^{\prime}))+\sum_{k=1}^{i}(-s-y^{\prime}\cdot\nabla\overline{\ell}^{k}_{h,i}(x^{\prime}))}{\sum_{k=1}^{i}(d^{k}_{h,i}+c^{k}_{h,i})}\right)\sum_{k=1}^{i}(d^{k}_{h,i}+c^{k}_{h,i})dx^{\prime}$

By (3.21), (6.22), (6.23) and (4.3) again, the last expression

$\displaystyle\leq$ $\displaystyle\frac{1}{2}\sum_{i=1}^{\infty}\int_{{\rm
int}[f]^{\prime}_{h,i}}\sum_{k=1}^{i}\phi\left((y^{\prime},t)\cdot\nabla
f(x^{\prime},\overline{\ell}^{k}_{h,i}(x^{\prime}))\right)c^{k}_{h,i}dx^{\prime}$
(6.25) $\displaystyle+\frac{1}{2}\sum_{i=1}^{\infty}\int_{{\rm
int}[f]^{\prime}_{h,i}}\sum_{k=1}^{i}\phi\left((y^{\prime},-s)\cdot\nabla
f(x^{\prime},-\underline{\ell}^{k}_{h,i}(x^{\prime}))\right)d^{k}_{h,i}dx^{\prime}$
$\displaystyle+\frac{1}{2}\sum_{i=1}^{\infty}\int_{{\rm
int}[f]^{\prime}_{h,i}}\sum_{k=1}^{i}\phi\left((y^{\prime},t)\cdot\nabla
f(x^{\prime},-\underline{\ell}^{k}_{h,i}(x^{\prime}))\right)d^{k}_{h,i}dx^{\prime}$
$\displaystyle+\frac{1}{2}\sum_{i=1}^{\infty}\int_{{\rm
int}[f]^{\prime}_{h,i}}\sum_{k=1}^{i}\phi\left((y^{\prime},-s)\cdot\nabla
f(x^{\prime},\overline{\ell}^{k}_{h,i}(x^{\prime}))\right)c^{k}_{h,i}dx^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\partial^{\ast}[f]_{h}/G_{1}}\phi\left((y^{\prime},t)\cdot\nabla
f(x)\right)|\nabla f(x)|^{-1}d\mathcal{H}^{n-1}(x)$
$\displaystyle+\frac{1}{2}\int_{\partial^{\ast}[f]_{h}/G_{1}}\phi\left((y^{\prime},-s)\cdot\nabla
f(x)\right)|\nabla f(x)|^{-1}d\mathcal{H}^{n-1}(x).$

Integrating both sides of the above inequality on $[0,\infty)$ with respect to
$h$, it follows from the co-area formula (4.14), Lemma 6.2, (6.14) and (6.15),
that

$\displaystyle\int_{SG}\phi\left((y^{\prime},t/2+s/2)\cdot\nabla
Sf(x)\right)dx$ (6.26) $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{G}\phi\left((y^{\prime},t)\cdot\nabla
f(x)\right)dx+\frac{1}{2}\int_{G}\phi\left((y^{\prime},-s)\cdot\nabla
f(x)\right)dx$ $\displaystyle=$ $\displaystyle 1.$

This and a glance at definition (5.1), gives (6.16), and thus (6.13) is
proved. ∎

Proof of Theorem 1.1. By Theorem 4.1, there exists a sequence of directions
$\\{u_{i}\\}$ such that the sequence defined by $f_{i+1}=S_{u_{i}}f_{i}$
(where $i=0,1,\dots$ and $f_{0}=f$) converges to $f^{\star}$ weakly in
$W^{1,1}$. By Lemma 5.5, there exists a convex body $K_{0}$ such that
$B_{\phi}(f_{i})$ converges to $K_{0}$ and $K_{0}\subset B_{\phi}(f^{\star})$.
Thus by Proposition 6.1,

$|B_{\phi}(f)|\leq|B_{\phi}(f_{1})|\leq\cdots\leq|B_{\phi}(f_{i})|\rightarrow|K_{0}|\leq|B_{\phi}(f^{\star})|.$

Thus, we have

$\mathcal{E}_{\phi}(f^{\star})\leq\mathcal{E}_{\phi}(f).$

∎

##  7 The affine Orlicz Pólya-Szegö principle and the Orlicz-Petty projection
inequality for star bodies

In [49], the Orlicz projection body $\Pi_{\phi}K$ of $K\in\mathcal{K}^{n}_{o}$
is defined as the body whose support function is given by

$\displaystyle h_{\Pi_{\phi}K}(u)=\inf\left\\{\lambda>0:\int_{\partial
K}\phi\left(\frac{u\cdot v_{K}(y)}{\lambda
h_{K}(v_{K}(y))}\right)h_{K}(v_{K}(y))d\mathcal{H}^{n-1}(y)\leq
n|K|\right\\}.$ (7.1)

Lutwak, Yang and Zhang [49] proved the following remarkable inequality.

Orlicz-Petty Projection inequality. Suppose $\phi\in\mathcal{N}$. If
$K\in\mathcal{K}^{n}_{o}$, then the volume ratio

$|\Pi^{\ast}_{\phi}K|/|K|$

is maximized when $K$ is an ellipsoid centered at the origin.

Zhang [63] gave the definition of the projection body for a compact set with
piecewise $C^{1}$. Similarly, for
$K\in\mathcal{S}_{o}^{n}\cap\mathcal{C}^{n}$, we define the Orlicz projection
body $\Pi_{\phi}K$ of $K$ as

$\displaystyle
h_{\Pi_{\phi}K}(u)=\inf\left\\{\lambda>0:\int_{\partial^{\ast}K}\phi\left(\frac{u\cdot
v_{K}(y)}{\lambda y\cdot v_{K}(y)}\right)y\cdot
v_{K}(y)d\mathcal{H}^{n-1}(y)\leq n|K|\right\\}.$ (7.2)

It is easily seen that $\Pi_{\phi}K$ is a convex body containing the origin in
its interior.

In this section, we shall prove that the affine Orlicz Pólya-Szegö principle
(1.5) implies the following Orlicz-Petty projection inequality for star
bodies.

Orlicz-Petty projection inequality for star bodies. Suppose
$\phi\in\mathcal{N}$. If $K\in\mathcal{S}^{n}_{o}\cap\mathcal{C}^{n}$, then
the volume ratio

$|\Pi^{\ast}_{\phi}K|/|K|$

is maximized when $K$ is an ellipsoid centered at the origin.

In (1.5), let

$\displaystyle f(x)=1-\|x\|_{K},\;\;\;x\in{\rm int}K,$ (7.3)

where $K\in\mathcal{S}_{o}^{n}\cap\mathcal{C}^{n}$. Since $[f]_{h}=(1-h)K$ is
a set of finite perimeter for any $h>0$ and $\int_{{\rm int}K}|\nabla
f|dx=\int_{0}^{1}P([f]_{h})dh<\infty$, $f\in W_{0}^{1,1}({\rm int}K)$. It is
easy to check that

$\displaystyle f^{\star}(x)=1-\|x\|_{K^{\star}},\;\;\;x\in{\rm int}K^{\star}.$
(7.4)

For $f$ as in (7.3), since $\phi(0)=0$ and $0<f\leq 1$, by (4.14), (3.11) and
(3.12), we have

$\displaystyle\frac{1}{|K|}\int_{{\rm intK}}\phi\left(\frac{u\cdot\nabla
f(x)}{\lambda}\right)dx$ (7.5) $\displaystyle=$
$\displaystyle\frac{1}{|K|}\int_{\\{x\in{\rm int}K,\nabla f(x)\neq
0\\}}\phi\left(\frac{u\cdot\nabla f(x)}{\lambda}\right)dx$ $\displaystyle=$
$\displaystyle\frac{1}{|K|}\int^{1}_{0}\int_{(1-h)\partial^{\ast}K}\phi\left(\frac{u\cdot\nabla
f(x)}{\lambda}\right)|\nabla f(x)|^{-1}d\mathcal{H}^{n-1}(x)dh$
$\displaystyle=$
$\displaystyle\frac{1}{|K|}\int^{1}_{0}\int_{\partial^{\ast}K}\phi\left(\frac{-u\cdot
v_{K}(y)}{\lambda y\cdot v_{K}(y)}\right)y\cdot
v_{K}(y)(1-h)^{n-1}d\mathcal{H}^{n-1}(y)dh$ $\displaystyle=$
$\displaystyle\frac{1}{n|K|}\int_{\partial^{\ast}K}\phi\left(\frac{-u\cdot
v_{K}(y)}{\lambda y\cdot v_{K}(y)}\right)y\cdot
v_{K}(y)d\mathcal{H}^{n-1}(y).$

By (5.1), (7.2) and (7.5), we have

$\displaystyle\|u\|_{f,\phi}$ $\displaystyle=$
$\displaystyle\inf\left\\{\lambda>0:\frac{1}{|K|}\int_{{\rm
int}K}\phi\left(\frac{u\cdot\nabla f(x)}{\lambda}\right)dx\leq 1\right\\}$
(7.6) $\displaystyle=$
$\displaystyle\inf\left\\{\lambda>0:\int_{\partial^{\ast}K}\phi\left(\frac{-u\cdot
v(y)}{\lambda y\cdot v(y)}\right)y\cdot v(y)d\mathcal{H}^{n-1}(y)\leq
n|K|\right\\}$ $\displaystyle=$ $\displaystyle h_{\Pi_{\phi}K}(-u).$

By (3.6), (7.6) and the definition of $B_{\phi}(f)$, we have

$\displaystyle B_{\phi}(f)=-\Pi^{\ast}_{\phi}K.$ (7.7)

By (7.7), $|K^{\ast}|=|K|$,
$\mathcal{E}_{\phi}(f^{\star})\leq\mathcal{E}_{\phi}(f)$ and
$\mathcal{E}_{\phi}(f)=|B_{\phi}(f)|^{-1/n}$, we have

$\displaystyle\frac{|\Pi^{\ast}_{\phi}K|}{|K|}=\frac{|B_{\phi}(f)|}{|K|}\leq\frac{|B_{\phi}(f^{\star})|}{|K^{\star}|}=\frac{|\Pi^{\ast}_{\phi}K^{\star}|}{|K^{\star}|},$
(7.8)

which is the Orlicz projection inequality for star bodies.

##  8 The Orlicz and affine Orlicz Pólya-Szegö inequalities

In this section, we shall prove that the affine Orlicz Pólya-Szegö inequality
(1.5) is essentially stronger than the Euclidean Orlicz Pólya-Szegö inequality
(1.3). Throughout this section, let $\phi\in\mathcal{N}$ be an even function.

######  Lemma 8.1.

For $f\in W_{0}^{n,\Phi}(G)$, then

$\displaystyle\frac{2\omega_{n-1}}{n\omega_{n}^{(n+1)/n}}\leq\frac{\mathcal{E}_{\phi}(f^{\star})}{\|\nabla
f^{\star}\|_{\Phi}}<\omega_{n}^{-\frac{1}{n}}.$ (8.1)

###### Proof.

For $v\in S^{n-1}$, since $\phi\in\mathcal{N}$ is an even function and by
Jensen’s inequality, we have

$\displaystyle\|v\|_{f^{\star},\phi}$ $\displaystyle=$
$\displaystyle\inf\left\\{\lambda>0:\int_{G^{\ast}}\phi\left(\frac{v\cdot\nabla
f^{\star}(x)}{\lambda}\right)dx\leq|G^{\star}|\right\\}$ (8.2)
$\displaystyle=$
$\displaystyle\inf\left\\{\lambda>0:\int^{D_{G^{\ast}}/2}_{0}\int_{S^{n-1}}\Phi\left(\frac{|v\cdot
u|\cdot|\nabla
f^{\star}(ru)|}{\lambda}\right)r^{n-1}dudr\leq|G^{\star}|\right\\}$
$\displaystyle\geq$
$\displaystyle\inf\left\\{\lambda>0:n\omega_{n}\int^{D_{G^{\ast}}/2}_{0}\Phi\left(\frac{1}{n\omega_{n}}\frac{\int_{S^{n-1}}|v\cdot
u|du}{\lambda}|\nabla f^{\star}(ru)|\right)r^{n-1}dr\leq|G^{\star}|\right\\}$
$\displaystyle=$
$\displaystyle\inf\left\\{\lambda>0:n\omega_{n}\int^{D_{G^{\ast}}/2}_{0}\Phi\left(\frac{1}{n\omega_{n}}\frac{2\omega_{n-1}}{\lambda}|\nabla
f^{\star}(ru)|\right)r^{n-1}dr\leq|G^{\star}|\right\\}$ $\displaystyle=$
$\displaystyle\inf\left\\{\lambda>0:\int^{D_{G^{\ast}}/2}_{0}\int_{S^{n-1}}\Phi\left(\frac{2\omega_{n-1}}{n\omega_{n}}\frac{|\nabla
f^{\star}(ru)|}{\lambda}\right)r^{n-1}dudr\leq|G^{\star}|\right\\}$
$\displaystyle=$ $\displaystyle\frac{2\omega_{n-1}}{n\omega_{n}}\|\nabla
f^{\star}\|_{\Phi}.$

Thus

$\displaystyle\|v\|_{f^{\star},\phi}\geq\frac{2\omega_{n-1}}{n\omega_{n}}\|\nabla
f^{\star}\|_{\Phi}.$ (8.3)

Because that $\phi$ is strictly increasing on $[0,\infty)$, thus by the second
equality of (8.2), for any $f\in W_{0}^{n,\Phi}(G)$,

$\displaystyle\|v\|_{f^{\star},\phi}<\|\nabla f^{\star}\|_{\Phi}.$ (8.4)

By (8.3) and (8.4), for any $f\in W_{0}^{n,\Phi}(G)$,

$\displaystyle\frac{2\omega_{n-1}}{n\omega_{n}}\leq\frac{\|v\|_{f^{\star},\phi}}{\|\nabla
f^{\star}\|_{\Phi}}<1.$ (8.5)

Combining (1.2) and (8.5) gives the desired inequality. ∎

######  Lemma 8.2.

If $f\in W_{0}^{n,\Phi}(G)$, then $\mathcal{E}_{\phi}(f)/\|\nabla f\|_{\Phi}$
is uniformly bounded from above by a positive constant, i.e.,

$\displaystyle\sup\left\\{\frac{\mathcal{E}_{\phi}(f)}{\|\nabla
f\|_{\Phi}}:\;\;f\in W_{0}^{n,\Phi}(G)\right\\}\leq\omega_{n}^{-\frac{1}{n}}.$
(8.6)

Moreover, $\mathcal{E}_{\phi}(f)/\|\nabla f\|_{\Phi}$ is not uniformly bounded
from below by any positive constant, i.e.,

$\displaystyle\inf\left\\{\frac{\mathcal{E}_{\phi}(f)}{\|\nabla
f\|_{\Phi}}:\;\;f\in W_{0}^{n,\Phi}(G)\right\\}=0.$ (8.7)

###### Proof.

On the one hand, since $\phi$ is even and strictly increasing on $[0,\infty)$,

$\displaystyle\|v\|_{f,\phi}$ $\displaystyle=$
$\displaystyle\inf\left\\{\lambda>0:\int_{G}\phi\left(\frac{|v\cdot u||\nabla
f(x)|}{\lambda}\right)dx\leq|G|\right\\}$ (8.8) $\displaystyle\leq$
$\displaystyle\inf\left\\{\lambda>0:\int_{G}\Phi\left(\frac{|\nabla
f(x)|}{\lambda}\right)dx\leq|G|\right\\}$ $\displaystyle=$
$\displaystyle\|\nabla f\|_{\Phi}.$

By (1.2) and (8.8)

$\mathcal{E}_{\phi}(f)=\left(\frac{1}{n}\int_{S^{n-1}}\|v\|^{-n}_{f,\phi}dv\right)^{-\frac{1}{n}}\leq\omega_{n}^{-\frac{1}{n}}\|\nabla
f\|_{\Phi}.$

Thus, (8.6) is established.

On the other hand, let $f_{1}(x)=f(Ax)$, where $A\in SL(n)$. By Lemma 5.2,
$\mathcal{E}_{\phi}(f)=\mathcal{E}_{\phi}(f_{1})$. Let

$A=\left(\begin{matrix}\tau&0&0&\cdots&0&\\\ 0&\frac{1}{\tau}&0&\cdots&0&\\\
0&0&1&\cdots&0&\\\ &&&\ddots&&\\\ 0&0&0&\cdots&1&\end{matrix}\right),$ (8.9)

where $\tau>0$. Let $D(f_{1})$ denote the domain of $f_{1}$. Then
$D(f_{1})=A^{-1}G$. Let $\eta=(\tau,1/\tau,1,\cdots,1)$ be a $n$-dimensional
vector, we have

$\displaystyle\|\nabla f_{1}\|_{\Phi}$ $\displaystyle=$
$\displaystyle\inf\left\\{\lambda>0:\;\frac{1}{|A^{-1}G|}\int_{A^{-1}G}\Phi\left(\frac{|A^{t}\nabla
f(Ax)|}{\lambda}\right)dx\leq 1\right\\}$ (8.10) $\displaystyle=$
$\displaystyle\inf\left\\{\lambda>0:\;\frac{1}{|G|}\int_{G}\Phi\left(\frac{|\eta\cdot\nabla
f(x^{\prime})|}{\lambda}\right)|A^{-1}|dx^{\prime}\leq 1\right\\}.$

Since $\phi$ is strictly monotone increasing on $(0,\infty)$, $\|\nabla
f_{1}\|_{\Phi}\rightarrow\infty$ when $\tau\rightarrow\infty$, which implies

$\inf\left\\{\frac{\mathcal{E}_{\phi}(f)}{\|\nabla f\|_{\Phi}}:\;\;f\in
W^{n,\Phi}(G)\right\\}=0.$

∎

By Theorem 1.1, Lemma 8.1 and Lemma 8.2, we have

$\displaystyle\frac{2\omega_{n-1}}{n\omega_{n}^{(n+1)/n}}\|\nabla
f^{\star}\|_{\Phi}\leq\mathcal{E}_{\phi}(f^{\star})\leq\mathcal{E}_{\phi}(f)\leq\omega^{-\frac{1}{n}}_{n}\|\nabla
f\|_{\Phi}.$ (8.11)

By (8.11), the following theorem is established.

######  Theorem 8.3.

For $f\in W_{0}^{n,\Phi}(G)$, if

$\mathcal{E}_{\phi}(f^{\star})\leq\mathcal{E}_{\phi}(f),$

then

$\|\nabla f^{\star}\|_{\Phi}\leq c_{1}\|\nabla f\|_{\Phi},$

where $c_{1}=\frac{n\omega_{n}}{2\omega_{n-1}}$ is a constant.

By Lemma 8.2 and Theorem 8.3, the affine Orlicz Pólya-Szegö inequality (1.5)
is essentially stronger than the Euclidean Orlicz Pólya-Szegö inequality
(1.3).

## References

  * [1] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000.
  * [2] M. Barchiesi, G.M. Capriani, N. Fusco, G. Pisante, Stability of Pólya-Szegö inequality for log-concave functions, J. Funct. Anal. 267 (2014), 2264-2297.
  * [3] G. Bianchi, R. J. Gardner, P. Gronchi, Symmetrization in geometry, Adv. Math. 306 (2017), 51-88.
  * [4] K.J. Böröczky, Stronger versions of the Orlicz-Petty projection inequality, J. Differential Geom. 95 (2013), 215-247.
  * [5] K.J. Böröczky, E. Lutwak, D. Yang, G. Zhang, The log-Brunn-Minkowski inequality, Adv. Math. 231 (2012), 1974-1997.
  * [6] J.E. Brothers, W.P. Ziemer, Minimal rearrangements of Sobolev functions, J. Reine Angew. Math. 384 (1988), 153-179.
  * [7] A. Burchard, Steiner symmetrization is continuous in $W^{1,p}$, Geom. Funct. Anal. 7 (1997), 823-860.
  * [8] A. Burchard, Y. Guo, Compactness via symmetrization, J. Funct. Anal. 214 (2004), 40-73.
  * [9] A. Burchard, A. Ferone, On the Extremals of the Pólya-Szegö Inequality, Indiana Univ. Math. J. 64 (2015), 1447-1463.
  * [10] G.M. Capriani, The Steiner rearrangement in any codimension, Calc. Var. Partial Differential Equations 49 (2014), 517-548.
  * [11] M. Chlebík, A. Cianchi, N. Fusco, The perimeter inequality under Steiner symmetrization: cases of equality, Ann. of Math. 162 (2005), 525-555.
  * [12] A. Cianchi, L. Pick, L. Slav ková Higher-order Sobolev embeddings and isoperimetric inequalities, Adv. Math. 273 (2015), 568-650.
  * [13] A. Cianchi, Second-order derivatives and rearrangements, Duke Math. J. 105 (2000), 355-385.
  * [14] A. Cianchi, L. Esposito, N. Fusco, C. Trombetti, A quantitative Pólya-Szegö principle, J. Reine Angew. Math. 614 (2008), 153-189.
  * [15] A. Cianchi, N. Fusco, Steiner symmetric extremals in Pólya-Szegö type inequalities, Adv. Math. 203 (2006), 673-728.
  * [16] A. Cianchi, N. Fusco, Minimal rearrangements, strict convexity and critical points, Appl. Anal. 85 (2006), 67-85.
  * [17] A. Cianchi, E. Lutwak, D. Yang, G. Zhang, Affine Moser-Trudinger and Morrey-Sobolev inequalities, Calc. Var. Partial Differential Equations 36 (2009), 419-436.
  * [18] A. Cianchi, On some aspects of the theory of Orlicz-Sobolev spaces. In Around the research of Vladimir Maz’ya. I, volume 11 of Int. Math. Ser. (N. Y.), 81-104. Springer, New York, 2010.
  * [19] L. Esposito, C. Trombetti, Convex symmetrization and Pólya-Szegö inequality, Nonlinear Anal. 56 (2004), 43-62.
  * [20] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Math., CRC Press, 1992.
  * [21] L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics 74, American Mathematical Society, Providence, RI, 1990.
  * [22] N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. 168 (2008), 941-980.
  * [23] R.J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002), 355-405.
  * [24] R.J. Gardner, Geometric Tomography, second edition, Cambridge University Press, New York, 2006.
  * [25] R.J. Gardner, D. Hug, W. Weil, The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities, J. Differential Geom. 97 (2014), 427-476.
  * [26] R.J. Gardner, D. Hug, W. Weil, D. Ye, The dual Orlicz-Brunn-Minkowski theory, J. Math. Anal. Appl. 430 (2015), 810-829.
  * [27] R.J. Gardner, G. Zhang, Affine inequalities and radial mean bodies, Amer. J. Math. 120 (1998), 505-528.
  * [28] P.M. Gruber, Convex and Discrete Geometry, Springer, Berlin, 2007.
  * [29] C. Haberl, E. Lutwak, D. Yang, G. Zhang, The even Orlicz Minkowski problem, Adv. Math. 224 (2010), 2485-2510.
  * [30] C. Haberl, F.E. Schuster, General $L_{p}$ affine isoperimetric inequalities, J. Differential Geom. 83 (2009), 1-26.
  * [31] C. Haberl, F.E. Schuster, Asymmetric affine $L_{p}$ Sobolev inequalities, J. Funct. Anal. 257 (2009), 641-658.
  * [32] C. Haberl, F.E. Schuster, J. Xiao, An asymmetric affine Pólya-Szegö principle, Math. Ann. 352 (2012), 517-542.
  * [33] C. Haberl, L. Parapatits, The centro-affine Hadwiger theorem, J. Amer. Math. Soc. 27 (2014), 685-705.
  * [34] J. Haddad, C.H. Jiménez, M. Montenegro, Sharp affine Sobolev type inequalities via the $L_{p}$ Busemann-Petty centroid inequality, J. Funct. Anal. 271 (2016), 454-473.
  * [35] A.-J. Li, D. Xi, G. Zhang, Volume inequalities of convex bodies from cosine transforms on Grassmann manifolds, Adv. Math. 304 (2017), 494-538.
  * [36] J. Li, G. Leng, Orlicz valuation, Indiana Univ. Math. J. 66 (2017), 791-819.
  * [37] Y. Lin, Smoothness of the Steiner symmetrization, to appear in Proc. Amer. Math. Soc. Available at http://dx.doi.org/10.1090/proc/13683.
  * [38] Y. Lin, Affine Orlicz Pólya-Szegö principle for log-concave functions, J. Funct. Anal. 273 (2017), 3295-3326.
  * [39] M. Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002), 158-168.
  * [40] M. Ludwig, Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), 4191-4213.
  * [41] M. Ludwig, Anisotropic fractional Sobolev norms, Adv. Math. 252 (2014), 150-157.
  * [42] M. Ludwig, M. Reitzner, A classification of $SL(n)$ invariant valuations, Ann. of Math. 172 (2010), 1219-1267.
  * [43] M. Ludwig, J. Xiao, G. Zhang, Sharp convex Lorentz-Sobolev inequalities, Math. Ann. 350 (2011), 169-197.
  * [44] E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131-150.
  * [45] E. Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), 244-294.
  * [46] E. Lutwak, D. Yang, G. Zhang, $L_{p}$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), 111-132.
  * [47] E. Lutwak, D. Yang, G. Zhang, Sharp affine $L_{p}$ Sobolev inequalities, J. Differential Geom. 62 (2002), 17-38.
  * [48] E. Lutwak, D. Yang, G. Zhang, Optimal Sobolev norms and the $L^{p}$ Minkowski problem, Int. Math. Res. Not. (2006), Art. ID 62987, 1-21.
  * [49] E. Lutwak, D. Yang, G. Zhang, Orlicz projection bodies, Adv. Math. 223 (2010), 220-242.
  * [50] E. Lutwak, D. Yang, G. Zhang, Orlicz centroid bodies, J. Differential Geom. 84 (2010), 365-387.
  * [51] V.G. Maz’ya, Sobolev Spaces, Springer-Verlag, Berlin (1985).
  * [52] V.G. Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer, Heidelberg, 2011.
  * [53] V.H. Nguyen, New approach to the affine Pólya-Szegö principle and the stability version of the affine Sobolev inequality, Adv. Math. 302 (2016), 1080-1110.
  * [54] G. Pólya, G. Szegö, Isoperimetric inequalities in mathematical physics, Ann. Math. Stud. 27, Princeton University Press (1951).
  * [55] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 151. Cambridge Univ. Press, Cambridge, 2014.
  * [56] G. Talenti, Inequalities in rearrangement invariant function spaces, In: M. Krbec, A. Kufner, B. Opic, J. Rákosnik (eds.) Nonlinear Analysis, Function Spaces and Applications, vol. 5, 177-230, Prometheus, Prague (1994).
  * [57] N.S. Trudinger, On new isoperimetric inequalities and symmetrization, J. reine angew. Math. 488 (1997), 203-220.
  * [58] T. Wang, The affine Sobolev-Zhang inequality on $BV(\mathbb{R}^{n})$, Adv. Math. 230 (2012), 2457-2473.
  * [59] T. Wang, The affine Pólya-Szegö principle: Equality cases and stability, J. Funct. Anal. 265 (2013), 1728-1748.
  * [60] E. Werner, D. Ye, New $L_{p}$ affine isoperimetric inequalities, Adv. Math. 218 (2008), 762-780.
  * [61] D. Xi, H. Jin, G. Leng, The Orlicz Brunn-Minkowski inequality, Adv. Math. 260 (2014), 350-374.
  * [62] D. Ye, Dual Orlicz-Brunn-Minkowski theory: dual Orlicz $L_{\phi}$ affine and geominimal surface areas. J. Math. Anal. Appl. 443 (2016), 352-371.
  * [63] G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), 183-202.
  * [64] B. Zhu, J. Zhou, W. Xu, Dual Orlicz-Brunn-Minkowski theory, Adv. Math. 264 (2014), 700-725.
  * [65] G. Zhu, The Orlicz centroid inequality for star bodies, Adv. in App. Math. 48 (2012), 432-445.
  * [66] D. Zou, G. Xiong, Orlicz-John ellipsoids, Adv. Math. 265 (2014), 132-168.
  * [67] W.P. Ziemer, Weakly Differentiable Functions: Sobolev spaces and functions of bounded variation, GTM 120, Springer Verlag, 1989.

School of Mathematics and Statistics, Chongqing Technology and Business
University,
Chongqing 400067, PR China
E-mail address<EMAIL_ADDRESS>
Department of Mathematics, Tandon School of Engineering, New York University,
6 MetroTech Center, Brooklyn, NY 11201, USA
E-mail address<EMAIL_ADDRESS>


# Development of Machine Vision Approach for Bolt Identification based on its
Dimension and Pitch for Mechanical Assembly Line

Toshit Jain1,a,∗, Faisel Mushtaq1,a, K Ramesh2,a, Sandip Deshmukh2,a,
Tathagata Ray1,a, Chandu Parimi3,a
Praveen Tandon4, Pramod Kumar Jha4 ∗Corresponding author Email addresses:
Toshit Jain<EMAIL_ADDRESS>Faisel Mushtaq:
<EMAIL_ADDRESS>Tathagata Ray<EMAIL_ADDRESS>1Department of Computer Science & Information Systems, BITS Pilani, Hyderabad
Campus  2Department of Mechanical Engineering, BITS Pilani, Hyderabad Campus
3Department of Civil Engineering, BITS Pilani, Hyderabad Campus  4Centre for
Advanced Systems, The Defence Research and Development Organisation  aBirla
Institute of Technology & Science, Pilani, Hyderabad Campus

###### Abstract

In this work, a highly customizable and scalable vision-based system for the
automation of classifying components(bolts in specific) in mechanical assembly
lines is described. The proposed system calculates the required features to
classify and identify the different kinds of bolts in the assembly line. The
system describes a novel method of calculating the pitch of the bolt in
addition to bolt identification and calculating the dimensions of the bolts.
Unlike machine learning-based systems, this identification and classification
system is extremely lightweight and can be run on bare minimum hardware such
as Raspberry Pi. The system is very fast in the order of milliseconds; hence
the system can be used successfully even if the components are steadily moving
on a conveyor. The results show that our system can correctly identify the
parts in our dataset with 98% accuracy using the calculated features.

###### Index Terms:

Bolts, Machine Vision, Image processing, Mechanical component identification,
Feature extraction

##  I Introduction

Whenever the production of mechanical items is concerned, assembly lines are
the most common production methods. Assembly lines include many small parts
like bolts, screws, fasteners, etc. Sorting them and classifying these parts
is a challenge. Historically this has been done manually, which is manpower
intensive and suffers manual errors. A small inaccuracy can result in
significant problems in quality and safety, especially in aerospace vehicle
assembly lines. These require strict quality assurance. Hence, introducing a
streamlined and automated process for this task holds merit and scope.

We are given various components for aerospace vehicle assembly. These
components include different types of bolts, fasteners, nuts, and washers.
Bolts have the highest count among all the components. Our objective is to
identify and classify bolts - quickly and accurately - from their image on the
assembly line. We also aim to provide different features relevant to the
assembly process, such as Diameter, Major Axis Length, Pitch, and the
Threading Type of the bolt. We are given thirty-three different kinds of bolts
that we aim to classify. The methodology discussed in this paper applies to
any type of bolt.

Figure 1: All 33 the Bolts used for this study

This paper solely relies on image processing and feature extraction to
identify the components, which has certain advantages over the already
existing machine learning algorithms [1, 2, 3]. This sole reliance on image
processing makes the system lightweight and quick to identify the components
without the added expense of training time on higher-end processors such as
GPUs. The proposed system can be run on a Raspberry Pi with fast execution
times, and the parts can be accurately identified even when moving swiftly on
a conveyor. In the case of the introduction of a new kind of bolt in the
assembly process, the proposed system is more resilient than the systems which
require retraining. This paper also introduces a novel method for calculating
the pitch of a bolt precisely using high-resolution cameras.

The paper is organized as follows: a summary of the previous related works are
presented in section II. A brief description of the dataset we use is given in
section III. The proposed methodology is discussed in section IV and the
results are shown in section V. Section VI concludes the findings of the
paper.

##  II Literature Review

The usage of mechanical components is common in assembly lines, yet the
research in automation of recognizing these components is still in its initial
stages. Identification of these mechanical components is an important phase
for ensuring the correctness of the assembly process. Among some related
works, Chávez et al. [4] proposed a Vision-Based Detection and Labelling of
Multiple Vehicle Parts to identify various parts in a car in ordinary scenes.
Their study is based on three steps a) Implementation of a machine vision
system to identify multiple car parts b) Introduction of a statistical model
for elimination of false detections and to infer the possible positions of
undetected parts. c) Introduction of geometrical models to model spatial
relationships among regions of interest and feasible search zones to delimit
search areas. Their system has achieved a precision of 95% in outdoor scenes.
A system for geometric feature inspection of mechanical parts [5] was
developed by Marshall et al. using laser range finder data was discussed in
1991. Among other significant works, Johan et al. [6] proposed a system for
recognition of bolt and nut based on backpropagation neural networks. Their
system has achieved recognition accuracy of 92% on a conveyor belt run at a
speed of 9 cm/sec. Similarly, Canali et al. [2] proposed a feature-based
automatic assembly parts detection and grasping system that can swiftly detect
parts and extract useful features by using contour mappings and feature
extraction. This work does not identify the parts but detects them with high
speed and accuracy without any machine learning methods. In another similar
work, Hao et al. (2015) [7] presented a machine vision system for
classification of machine parts in real-time by using an embedded image card
with Field Programming Gate Array (FPGA) for accelerating the computation.
Similarly, Guo et al. [8] also proposed a method for material sorting system
based on machine vision that relied heavily on image preprocessing and edge
detection but was only effective for a few predefined shapes. In other similar
works, Huang et al. [9] proposed component assembly inspection based on mask
R-CNN and support vector machines. Also, Killing et al. (2009) [10] developed
a system for the detection of missing fasteners on steel stampings based on
neuro-fuzzy and threshold-based picture classification algorithms. The authors
conclude that neuro-fuzzy has a far lower RMS error than the threshold-based
classification algorithm.

##  III Setup and Dataset

Figure 2: The Setup

An image acquisition system has been developed to identify bolts for capturing
quality images. An opal-frosted acrylic sheet has been used to fabricate the
testbed base, supported by a backlit LED source with adjustable light
intensity to eliminate shadow and reflection. To capture images, a 4MP
monochrome industrial camera has been mounted above the testbed at a fixed
position to capture the lateral view of bolts. This lateral view (as shown in
Fig. 3) of the components is eventually used to determine the dimension and
pitch of the components. The setup is also enclosed with black acrylic sheets
to block unnecessary light from the surroundings. Since the captured images
are the silhouettes of the bolts, they are easy to convert to a binary format
using thresholding. We have used the OpenCV library[11] `cv2.threshold()` to
get the binary equivalent of the image. A binary image is essentially one with
just two values: 0 and 1, where value 0 represents black and value 1
represents white. An example of this is shown in Fig. 4. Also, since the
distance between the camera and the testbed is always constant, we can easily
convert the pixels into millimeters by multiplying them with a conversion
factor (pixels per mm) to compensate for perspective shortening.

Figure 3: Sample image of a top view in the dataset Figure 4: Binary Image of
a full thread bolt

##  IV Methodology

A bolt is kept inside the setup, and the top camera captures an image. The
bolt is identified by three features, namely, the length of the major axis,
the length of the minor axis, and the type of threading. Pitch is also an
important feature that we calculate. The other features we focus on are area
and perimeter. These two features aren’t physically significant, but we use
them to calculate other features. The features are defined as follows:

  * •

Length of major axis: This is defined as the full lateral length of the bolt.
See Fig. 4

  * •

Length of minor axis: This is the diameter of the body of the bolt. See Fig. 4

  * •

Type of threading: We have two types of bolts in our dataset: Full thread
bolts and half thread bolts. Fig. 6 shows the silhouette of a full thread bolt
and Fig. 9 shows the silhouette of a half thread bolt.

  * •

Pitch: Pitch is the distance between two consecutive threads. See Fig. 4

  * •

Area: This is the area of the lateral cross-section of a bolt. The total
number of white pixels in Fig. 6 is the area of the bolt.

  * •

Perimeter: This is the perimeter of the shape that the silhouette of the bolt
makes. The total number of white pixels in Fig. 5 is the perimeter of the
bolt.

###  IV-A Getting the region of interest in minimum possible area

In a binary image, we only have two types of pixels, black and white, with
values 0 and 255 respectively. Getting an upright bounding box is fairly
straightforward since the pixels for a bolt form connected components in a
graph. The top, right, bottom, and left bounds are determined by the topmost,
rightmost, bottom-most, and leftmost white pixels. The `cv2.boundingRect()`
method can be used to get this bounding box. Similarly, edge detection can
also be done by using `cv2.findContours()`. We find a rectangle, possibly
rotated, with the minimum area that encloses the whole component. This is
easily implemented by first finding the contours of the bolt by using
`cv2.findContours()` method from the OpenCV Library which basically stores the
shape of the component and then we get the minimum area rectangle by calling
`cv2.minAreaRect()`.

Figure 5: Contour of a bolt

###  IV-B Calculation of area and perimeter

The area of a component in a binary image is the number of white pixels since
the white pixels represent the cross-section of the bolt. The area calculation
is very helpful in orienting the image, getting the type of threading, and
calculating the pitch. Perimeter is calculated very easily using
`cv2.arcLength()` on the contour of the screw since we already have calculated
the contours in section IV-A. The contour of a screw looks as shown in Fig. 5.
The perimeter helps identify the threading type.

###  IV-C Aligning and orienting the minimum area rectangle

We align the minimum area rectangle along the x and y axes obtained in section
IV-A by using `cv2.getPerspectiveTransform()` and warping this perspective to
an upright rectangle that is aligned with the axes, with the same height and
width that was returned by `cv2.minAreaRect()` in section IV-A by using the
`cv2.warpPerspective()` method. Then we rotate the aligned rectangle such that
the length of the rectangle is longer than its height. This length is the
length of the major axis.

As a convention, we keep the head of the bolt on the left side of the
rectangle. Since our rectangle is already laid out horizontally, we need to
determine if the head is on the left or right side. For this, we cut the image
from the middle and divided it into two parts. We calculate the areas in both
the image’s left and right half. Since the image is binary, the area is the
total number of white pixels. The head is on the side where we have a greater
value of the area. If the area of the right half of the image is larger, then
we rotate the image 180 degrees. The image after orienting is shown in Fig. 6.

Figure 6: The aligned and oriented minimum area rectangle

###  IV-D Returning the major and minor axes

The length of the minimum area rectangle is the length of the major axis, in
pixels. To get the length of the minor axis, we cut the oriented image in half
and drew a minimum area rectangle on the right half. The width of this new
rectangle is the length of minor axis, d, in pixels.

###  IV-E Removing the head

We have an oriented binary image of the bolt with its head on the left. In all
the types of bolts in our dataset, along the length of the bolt, the head does
not take more than 20% of the length of the bolt. The algorithm to remove the
head is similar to binary search. Suppose the length of the image, i.e. the
length of the bolt is l, and the width of the image, i.e. the width of the
head of the bolt is w. We also already have the length of the minor axis, d An
element for our “binary search” algorithm is defined as a one-pixel wide and w
pixels long strip of our image. The starting element is the leftmost strip of
the image and the last element is the strip at 0.2l from the left end of the
image.

In every step of the search, we slice the image from the middle point, m to
the end of the image, l, and get a minimum area rectangle of this sliced
image. If the width of this minimum area rectangle, w’ is approximately the
same as w, then we repeat this in the right half. If w’ is approximately the
same as d then that means the entire head is to the left of m and we search
again in the left half. We stop when we find an element h where the width of
the minimum area rectangle from 0 to h-1 is approximately w and the width of
the minimum area rectangle from h to l is approximately d. We finally return
an image slice from h to l which corresponds to the body of the bolt.

Note that if the head of the bolt is required then we can return the image
slice from 0 to h-1.

Algorithm 1 gives this pseudocode in detail. The variables used are visually
depicted in Fig. 7 It uses a parameter $thresh$. This parameter can be set to
0 to get the exact cutting point. In practice, we set this parameter to 5.
Though this does not provide the exact cutting point and actually takes away 5
extra strips from the head side of the bolt, this is practically better for
the algorithm. This parameter’s significance is explained more in section V.
The output of this algorithm is shown in Fig. 8

Figure 7: Variables used in algorithm 1 Algorithm 1 Algorithm for removing
head from a bolt

0: Oriented minimum area rectangle image img of the bolt

0: Image of the bolt without the head of the bolt

1: $l\leftarrow length(img)$

2: $w\leftarrow width(img)$

3: $d\leftarrow diameterOfBolt$

4: $lowerBound\leftarrow 0$

5: $upperBound\leftarrow 0.2l$

6: $thresh\leftarrow 5$

7: while True do

8: $mid\leftarrow(lowerBound+upperBound)/2$

9: $temp\leftarrow minAreaRect(img[mid:l])$

10: $w^{\prime}\leftarrow width(temp)$

11: if ($abs(w^{\prime}-w)\leq abs(w^{\prime}-d)$) then

12: $upperBound\leftarrow mid+1$

13: else

14: $lowerBound\leftarrow mid-1$

15: $\\_temp\leftarrow minAreaRect(img[mid-1:l])$

16: $w^{\prime\prime}\leftarrow width(\\_temp)$

17: if ($abs(w^{\prime\prime}-w)\leq abs(w^{\prime\prime}-d)$) then

18: return $minAreaRect(img[mid+thresh:l])$

19: end if

20: end if

21: end while

22: return $minAreaRect(img[mid+thresh:l])$

Figure 8: Bolt after it’s head is removed (Output of algorithm 1)

###  IV-F Getting the threading type

The labeling method that we use to identify a bolt uses three labels: length
of the bolt, diameter, and type of threading. Generally, there are two types
of threading in the bolts:

  * •

Full thread, in which the threading goes all the way from the bottom to the
head along the body of the bolt.

  * •

Half thread, in which the threading starts from the bottom to about 35-40% of
the length of the bolt.

To identify the type of threading, we start with the headless image of the
bolt obtained in section IV-E. We divide the image into two parts by cutting
it from the middle. Now we compare the two halves to determine if the bolt is
half threaded. In a half-threaded bolt, the entire threaded part lies in the
right half. We calculate the contours of both halves using the
`cv2.findContours()` method in the OpenCV library. If the bolt is half
threaded, then the contour of the left half of the bolt is basically a
rectangle, which is a convex polygon. So if the `cv2.isContourConvex()`
function returns True on the left contour, then we conclude the bolt is half
threaded. If it returns False then we must run some other tests before
concluding the bolt to be fully threaded because some noise in the image could
have resulted in minor irregularities in the contour that may have resulted it
to test negative on the convex polygon check. Then we compare the areas and
the perimeters of the two halves. If the perimeter of the right half is
considerably larger than that of the left half, then we can conclude that the
bolt is half threaded because a fully threaded bolt would have similar
perimeters in both halves. Otherwise, we check if the area of the left half is
comparable to the area of the whole left part of the image. If this turns out
to be true then we can say that the bolt is half threaded. If all the three
tests turn out negative only then we say that the bolt is a full thread bolt.

Figure 9: Binary image of a half thread bolt

###  IV-G Identification of the bolt

The actual size of a bolt differs from the size that is indicated by the name
of the bolt. For example, the bolt indicated by M5X12_FT has a diameter equal
to 4.90 mm and a length equal to 12.06 mm. Moreover, a slight variation in the
dimensions is observed when the bolt is kept in the middle of the setup, i.e.
directly below the camera, as compared to when it is kept in the corner. This
is due to perspective shortening. If we directly convert the lengths of the
axes that we obtained in section IV-D, from pixels to millimeters, then we
must keep these two shortcomings in mind. To overcome these two obstacles, and
simplify the problems, we adopt a different approach for bolt identification.
We create a minimum area rectangle of all bolts and store them in a lookup
table. These minimum area rectangles are manually verified to be correct and
precise. Then, their widths and heights, in pixels, are stored in tabular form
in a lookup table. Then, our method calculates the dimensions and the
threading type, and the closest match from the lookup table is returned. We
use Euclidean distance to calculate the closest match.

###  IV-H Calculation of pitch

Pitch is the distance between two consecutive threads. To calculate pitch, we
take the headless image of the bolt obtained in section IV-E. Regardless of
the bolt’s threading type, 30% of the screw from the right will always be
threaded. We slice out this part of the bolt. We’ll calculate the pitch on
this part of the image. First, we construct a minimum area rectangle on this
image and align and orient it horizontally as in section IV-C. Next, we
calculate its contour and generate its perimeter shape. We also generate the
shape of the minimum area rectangle. Ideally, all the crests should touch the
upper side of the rectangle. We could calculate the points of intersection of
the two contours, and we should be able to calculate pitch. But there may be
minor irregularities due to noise, image resolution, blur, etc., and some
crests may end up a pixel or two below the upper side of the rectangle.
Therefore to be sure that all the crests intersect the rectangle, we nudge the
upper side down by some pixels. Therefore for every crest, we get two
intersection points. Next, we do an AND operation of the two contours. We
traverse the length of the upper side and count the number of white pixels we
encounter. Let the total number of white pixels encountered be n. We also
store the location of the first white pixel, a that we encounter, and the last
white pixel, b we encounter. Subtracting these two gives the stretch length in
which we counted the pixels. The total number of white pixels is half of our
actual count. Therefore, the average pitch value is given by:

$pitch=\frac{b-a}{\frac{n}{2}}$ (1)

##  V Results

###  V-A Perspective Shortening

One of the innate challenges of the setup is perspective shortening. The
lengths of the bolt as calculated in the corner can differ from the lengths as
calculated in the center by 1.4% in the worst case. The conversion factor is
calculated for the center of the setup and gives more accurate results when
the bolt is kept in the center of the setup. For identification, the results
are the same irrespective of where the bolt is kept. This is because an error
of 1.4% does not change the class of the bolt. No two components have
dimensions within 1.4% of each other. But this error does become an issue for
pitch calculation. Therefore, all the pitch calculations are made with the
bolt kept in the center for more accurate results. The differences in
measurements for some bolts are shown in table I. From this table, it is
apparent that the bolt measurements (in pixels) appear to be larger when the
bolt is kept in the center as compared to when it is kept in the corner in
different orientations. The difference is observed to be maximum when the bolt
is kept in the extreme corner parallel to the side of the testbed.

TABLE I: Measurements of Bolts when kept at the corners and the center, in pixels Bolt | Center | Corner | Percentage
---|---|---|---
Name | Height | Width | Height | Width | Error
M8x35_HT | 147 | 407 | 147 | 411 | 0.98%
M10x50_HT | 181 | 577 | 183 | 579 | 1.22%
M10x35_FT | 179 | 426 | 181 | 428 | 1.21%
M4x75_FT | 75 | 781 | 75 | 783 | 0.26%

###  V-B Identification of the Bolts

Figure 10: Differences between the experimental and actual measurements

The major and minor axes of all the bolts were measured. The conversion factor
used was 12.42 pixels per millimeter. Both in Fig. 10 and Fig. 11, the bolts
are arranged on the x-axis in increasing order of the lengths of the bolts.
The difference in the measurements (in millimeters) is shown in Fig. 10. The
measurements of the minor axis are very close to the actual values. We can see
a larger difference in measurements of the major axis as the length of the
bolts increases. But as seen in Fig. 11 the actual percentage error is very
less as the length of the bolt is very large. For example, the length of the
last bolt was measured to be 72.70mm while the actual length was 75.43 mm.
This error of 2.73mm on 75.43mm bolts is a very low 2.67% of error. In Fig. 11
the percentage errors in measurements are calculated after rounding off to the
nearest integer. There is a peak in percentage errors in minor axis
measurement. The M8x20_FT bolt was measured to have 8.61 mm minor axis. This
resulted in it to getting rounded off to 9mm and this 1 mm error resulted in a
12.5% error. But eventually, after taking the major axis into account, the
bolt will get classified into the correct class. This is because we use
Euclidean distance for identification.

Figure 11: Percentage errors of bolt measurements after rounding off to
nearest whole number

Fig. 12 shows the results of an experiment on 200 bolts. 5 to 8 bolts were
placed on the testbed with random orientation. The results were produced by
comparing the Euclidean distance of the height and width of the minimum area
rectangles with the template table as discussed in section IV-G. The first
graph in Fig. 12 shows that the dimensions of all the 200 bolts were correctly
identified with 100% accuracy. The true positive curve exactly follows the
total number of components curve. The false positives are zero. The second
graph in Fig. 12 shows the results of the identification of the type of
threading. A total of 9 false positives were detected. All 9 false positives
were half-threaded bolts that were incorrectly identified as full-threaded
bolts. The accuracy for identification of threading type remains 95.5%. The
full-threaded bolts were all correctly identified as full-threaded with 100%
accuracy. In the third graph in Fig. 12 the results of the identification of
the whole bolts are shown. The false positives drop to just 4. This is because
if we get a bolt that matches only one entry in the lookup table then the
threading type is not needed for the full identification of the bolt. For
example, we can see that the M5x25_HT bolt was incorrectly identified as full-
threaded 3 out of 7 times. But these 3 false positives drop to zero as we have
no full threaded bolt with the same dimensions. Hence we prefer the
dimensional results and ignore the threading result in these cases. The
accuracy we get for the identification of the whole bolt is 98% on a test set
of 200 images.

Figure 12: True positives and false positives for identification of different
parameters of a bolt over a set of 200 images

###  V-C Head Removal

In section IV-E the parameter thresh was used to control the positions of the
cut point for the removal of the head from a bolt. When the bolt is kept in
the center of the testbed then the value of thresh can be kept as 0 and that
works perfectly. When the bolt is kept in the corner, the head removed image
can look like Fig. 13. Therefore we keep the value of thresh as 5 for these
corner cases.

Figure 13: Result of algorithm 1 with thresh values 1 and 5

###  V-D Pitch Calculation

The camera used in our setup is 4MP. This resolution limits the amount of
details captured. Because of this limitation, the pitch calculation was done
only on larger components, and the component was placed in the center of the
testbed, directly below the camera. Even though the camera was of low
resolution, we were able to get the pitch between $\pm$ 0.07 mm of the actual
pitch for all the bolts that we calculated pitch for. This can easily be made
more accurate by using cameras with higher resolution or by reducing the
distance between the camera and the testbed.

###  V-E Performance

The proposed system has shown good performance in identifying different bolts.
The system was implemented in Python and run on a) a Macbook with an Intel i5
at 2.30GHz, 8192MiB RAM, and macOS 12, and b) a Raspberry Pi with a quad-core
BCM2711 at 1.500GHz, 1872MiB RAM, and Raspbian GNU/Linux 10. Table II
summarizes the execution times of the algorithms in the task of getting all
the features, namely, Pitch Calculation, Calculation of the Major and Minor
Axes, Threading type Identification, and Classifying the Bolt.

TABLE II: Performance of the System Average time of execution
---
Macbook | 105 ms
Raspberry Pi | 233 ms

##  VI Conclusion

In this paper, a system for the identification of various features for various
bolts was presented. The system has immediate applications in industrial
assembly lines that contain a lot of bolts. The system describes a way to get
the type of threading, dimensions, and pitch of a bolt. From the experimental
results, the method discussed guarantees the successful identification of
bolts. The model was 100% successful in calculating the lengths of the major
and minor axes. This accurate calculation of the dimension was also useful in
sorting out defective and/or wrong parts. The accuracy of the model was found
to be 98% for the correct identification of the bolts. The identification of
bolts that were larger than 40 mm was 100% accurate. Hence, the accuracy of
the model can be improved to 100% with higher resolution cameras. The pitch
calculation is close to accurate on larger parts but is unreliable for smaller
bolts in the current setup due to hardware limitations. Currently, the pitch
can be calculated within an error range of $\pm$0.07 mm. The system is fast
and, even with limited computing resources, can successfully identify
components on a swiftly moving conveyor belt.

In future works, we intend to define methods for the classification of
different components like nuts, bolts, washers, and other fasteners using
computer vision, and then propose similar image processing methods for
extracting the features of these classified components.

## Acknowledgments

The authors gratefully acknowledge ARMREB, New Delhi for providing financial
support to carry out this research (Sanction Letter No.:
ARMREB/ADMB/2020/224). The authors are thankful to Birla Institute of
Technology & Science, Pilani, Hyderabad Campus for their support in carrying
out this work.

## References

  * [1] R. Mehrotra, F. K. Kung, and W. I. Grosky, “Industrial part recognition using a component-index,” _Image and Vision Computing_ , vol. 8, no. 3, pp. 225–232, 1990.
  * [2] C. Canali, F. Cannella, F. Chen, G. Sofia, A. Eytan, and D. G. Caldwell, “An automatic assembly parts detection and grasping system for industrial manufacturing,” in _2014 IEEE International Conference on Automation Science and Engineering (CASE)_ , 2014, pp. 215–220.
  * [3] J. W. Perng, Y. W. Hsu, Y. Z. Yang, C. Y. Chen, and T. K. Yin, “Development of an embedded road boundary detection system based on deep learning,” _Image and Vision Computing_ , vol. 100, p. 103935, 2020. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0262885620300676
  * [4] A. Chávez-Aragón, R. Laganière, and P. Payeur, “Vision-based detection and labelling of multiple vehicle parts,” in _2011 14th International IEEE Conference on Intelligent Transportation Systems (ITSC)_ , 2011, pp. 1273–1278.
  * [5] A. Marshall, R. Martin, and D. Hutber, “Automatic inspection of mechanical parts using geometric models and laser range finder data,” _Image and Vision Computing_ , vol. 9, no. 6, pp. 385–405, 1991. [Online]. Available: https://www.sciencedirect.com/science/article/pii/026288569190005A
  * [6] T. M. Johan and A. S. Prabuwono, “Recognition of bolt and nut using artificial neural network,” in _2011 International Conference on Pattern Analysis and Intelligence Robotics_ , vol. 1, 2011, pp. 165–170.
  * [7] H. Guo, H. Xiao, S. Wang, W. He, and K. Yuan, “Real-time detection and classification of machine parts with embedded system for industrial robot grasping,” in _2015 IEEE International Conference on Mechatronics and Automation (ICMA)_ , 2015, pp. 1691–1696.
  * [8] C. L. Guo, H. Zhu, Y. Ma, X. Xiao, and K. Sun, “The research of material sorting system based on machine vision,” in _2020 IEEE 4th Information Technology, Networking, Electronic and Automation Control Conference (ITNEC)_ , vol. 1, 2020, pp. 1840–1843.
  * [9] H. Huang, Z. Wei, and L. Yao, “A novel approach to component assembly inspection based on mask r-cnn and support vector machines,” _Information_ , vol. 10, no. 9, p. 282, 2019.
  * [10] J. Killing, B. Surgenor, and C. Mechefske, “A machine vision system for the detection of missing fasteners on steel stampings,” _The International Journal of Advanced Manufacturing Technology_ , vol. 41, no. 7-8, p. 808, 2009\.
  * [11] G. Bradski, “The OpenCV Library,” _Dr. Dobb’s Journal of Software Tools_ , 2000.