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We provide an overview of projection bodies in Gaussian probability space for sets of finite Gaussian perimeter and their corresponding applications in functions of Bounded variation space. On the one hand, we study the properties of Gaussian projection bodies for sets of finite Gaussian perimeter under Ehrhard symmetrization and establish a Gaussian projection-type inequality. This inequality concludes that Ehrhard symmetrization contracts the Minkowski symmetrization of the Gaussian projection body for set of finite Gaussian perimeter |$E$|⁠. On the other hand, we investigate the functional "lifting" of Ehrhard symmetrization and establish the affine Gaussian Pólya–Szegö-type inequalities in terms of the functional Ehrhard symmetrization. [ABSTRACT FROM AUTHOR]

In Lin (Calc. Var. Partial Differential Equations 60:196, 2021), the first author proved the Steiner's projection inequality and the Petty projection inequality for sets of finite perimeter but didn't characterize their cases of equality. In this paper, we characterize the case of equality in the Steiner's projection inequality. Moreover, we give a new proof of the equality case in the Petty projection inequality, which was characterized by Wang (Adv. Math. 230:2457–2473, 2012). [ABSTRACT FROM AUTHOR]

The Petty projection inequality for sets of finite perimeter is proved. Our approach is based on Steiner symmetrization. Neither the affine Sobolev inequality nor the functional Minkowski problem is used in our proof. Moreover, for sets of finite perimeter, we prove the Petty projection inequality with respect to Steiner symmetrization. [ABSTRACT FROM AUTHOR]

The conjecture about the Orlicz Pólya–Szegö principle posed in [ 42 ] is proved. The cases of equality are characterized in the affine Orlicz Pólya–Szegö principle with respect to Steiner symmetrization and Schwarz spherical symmetrization. [ABSTRACT FROM AUTHOR]

The affine L p Pólya–Szegö principle significantly strengthens the classical Pólya–Szegö principle. It is an open problem whether there exists an affine Orlicz Pólya–Szegö principle which includes the affine L p Pólya–Szegö principle as special case. In this paper, an affine Orlicz Pólya–Szegö principle for log-concave functions is established by using functional Steiner symmetrizations. [ABSTRACT FROM AUTHOR]

For sets of finite perimeter, k -codimensional projection bodies are introduced. The corresponding k -codimensional projection inequalities are investigated and their equality cases are characterized. [ABSTRACT FROM AUTHOR]

The original goal of this paper is to extend the affine isoperimetric inequality and Steiner type inequality of Orlicz projection bodies (which originated to Lutwak, Yang, and Zhang [32]), from convex bodies to Lipschitz star bodies (whose radial functions are locally Lipschitz). In order to achieve it, we investigate the graph functions of the given Lipschitz star body K : Along almost all directions u , we can define the graph functions on an open dense subset of the orthogonal projection of K onto u ⊥. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we extend the filled function method proposed in Lin et al. (2009) [10] to solve the constrained system of nonlinear equations, and we propose two methods to find a solution of such a system. Finally the efficiency of the present method is illustrated through some examples. [Copyright &y& Elsevier]

Abstract: A new proof of the Mahler conjecture in is given. In order to prove the result, we introduce a new method — the vertex removal method; i.e., for any origin-symmetric polygon , there exists a linear image contained in the unit disk , and there exist three contiguous vertices of lying on the boundary of . We can show that the volume-product of decreases when we remove the middle vertex of the three vertices. [ABSTRACT FROM AUTHOR]

Abstract: Systems of nonlinear equations are ubiquitous in engineering, physics and mechanics, and have myriad applications. Generally, they are very difficult to solve. In this paper, we will present a filled function method to solve nonlinear systems. We will first convert the nonlinear systems into equivalent global optimization problems with the property: is a global minimizer if and only if its function value is zero. A filled function method is proposed to solve the converted global optimization problem. Numerical examples are presented to illustrate our new techniques. [Copyright &y& Elsevier]

Abstract: In this paper, a new global optimization approach based on the filled function method is proposed for solving box-constrained systems of nonlinear equations. We first convert the nonlinear system into an equivalent global optimization problem, and then propose a new filled function method to solve the converted global optimization problem. Several numerical examples are presented and solved by using different local minimization methods, which illustrate the efficiency of the present approach. [Copyright &y& Elsevier]

The affine Orlicz Pólya–Szegö principles with respect to Steiner symmetrization and Schwarz symmetrization proved in Lin and Xing (International mathematics research notices, 2019) are extended to B V (Ω) , the space of functions of bounded variation defined in an open set Ω ⊂ R n , and their equality cases are characterized. [ABSTRACT FROM AUTHOR]