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25 February 2025, Volume 45 Issue 1 Previous Issue    Next Issue

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PREFACE Acta mathematica scientia,Series B. 2025, 45 (1):  1-2.  DOI: 10.1007/s10473-025-0100-3 Abstract ( 119 )   Save Related Articles | Metrics

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$\Gamma$-CONVEXITY Zhouqin Jia, Wenzhi Liu Liping Yuan, Tudor Zamfirescu Acta mathematica scientia,Series B. 2025, 45 (1):  3-15.  DOI: 10.1007/s10473-025-0101-2 Abstract ( 160 )   Save Let $\mathcal{F}$ be a family of sets in $\mathbb{R}^d\ (\mathrm{always}\ d\geq 2)$. A set $M\subset\mathbb{R}^d$ is called $\mathcal{F}$-convex, if for any pair of distinct points $x, y \in M$, there is a set $F\in \mathcal{F}$ such that $x, y \in F$ and $F \subset M$. We obtain the $\Gamma$-convexity, when $\mathcal{F}$ consists of $\Gamma$-paths. A $\Gamma$-path is the union of both shorter sides of an isosceles right triangle. In this paper we first characterize some $\Gamma$-convex sets, bounded or unbounded, including triangles, regular polygons, subsets of balls, right cylinders and cones, unbounded planar closed convex sets, etc. Then, we investigate the $\Gamma$-starshaped sets, and provide some conditions for a fan, a spherical sector and a right cylinder to be $\Gamma$-starshaped. Finally, we study the $\Gamma$-triple-convexity, which is a discrete generalization of $\Gamma$-convexity, and provide characterizations for all the 4-point sets, some 5-point sets and $\mathbb{Z}^{d}$ to be $\Gamma$-triple-convex. References | Related Articles | Metrics

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NOTES ON THE LOG-MINKOWSKI INEQUALITY OF CURVATURE ENTROPY Deyi LI, Lei MA, Chunna ZENG Acta mathematica scientia,Series B. 2025, 45 (1):  16-26.  DOI: 10.1007/s10473-025-0102-1 Abstract ( 245 )   Save An upper estimate of the new curvature entropy is provided, via the integral inequality of a concave function. For two origin-symmetric convex bodies in $\mathbb{R}^n$, this bound is sharper than the log-Minkowski inequality of curvature entropy. As its application, a novel proof of the log-Minkowski inequality of curvature entropy in the plane is given. References | Related Articles | Metrics

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MINIMAL WIDTHS AND ORTHOGONALITY TYPES Chan He, Horst Martini, Senlin Wu Acta mathematica scientia,Series B. 2025, 45 (1):  27-39.  DOI: 10.1007/s10473-025-0103-0 Abstract ( 110 )   Save The minimal widths of three bounded subsets of the unit sphere associated to a unit vector in a normed linear space are studied, and three related geometric constants are introduced. New characterizations of inner product spaces are also presented. From the perspective of minimal width, strong $\varepsilon$-symmetry of Birkhoff orthogonality is introduced, and its relation to $\varepsilon$-symmetry of Birkhoff orthogonality is shown. Unlike most of the existing parameters of the underlying space, these new constants are full dimensional in nature. References | Related Articles | Metrics

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WEIGHTED CONE-VOLUME MEASURES OF PSEUDO-CONES Rolf Schneider Acta mathematica scientia,Series B. 2025, 45 (1):  40-51.  DOI: 10.1007/s10473-025-0104-z Abstract ( 109 )   Save A pseudo-cone in $\mathbb{R}^{n}$ is a nonempty closed convex set $K$ not containing the origin and such that $\lambda K \subseteq K$ for all $\lambda\ge 1$. It is called a $C$-pseudo-cone if $C$ is its recession cone, where $C$ is a pointed closed convex cone with interior points. The cone-volume measure of a pseudo-cone can be defined similarly as for convex bodies, but it may be infinite. After proving a necessary condition for cone-volume measures of $C$-pseudo-cones, we introduce suitable weights for cone-volume measures, yielding finite measures. Then we provide a necessary and sufficient condition for a Borel measure on the unit sphere to be the weighted cone-volume measure of some $C$-pseudo-cone. References | Related Articles | Metrics

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A FUNCTIONAL ORLICZ BUSEMANN-PETTY CENTROID INEQUALITY FOR LOG-CONCAVE FUNCTIONS Xiao Li, Jiazu Zhou Acta mathematica scientia,Series B. 2025, 45 (1):  52-71.  DOI: 10.1007/s10473-025-0105-y Abstract ( 125 )   Save In this paper, the Orlicz centroid function for log-concave functions is introduced. A rearrangement inequality of the Orlicz centroid function for log-concave functions is obtained. The rearrangement inequality implies the Orlicz Busemann-Petty centroid inequality of Lutwak, Yang and Zhang [23]. References | Related Articles | Metrics

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ON GENERALIZED KISSING NUMBERS OF CONVEX BODIES Yiming Li, Chuanming Zong Acta mathematica scientia,Series B. 2025, 45 (1):  72-95.  DOI: 10.1007/s10473-025-0106-x Abstract ( 86 )   Save In 1694, Gregory and Newton proposed the problem to determine the kissing number of a rigid material ball. This problem and its higher dimensional generalization have been studied by many mathematicians, including Minkowski, van der Waerden, Hadwiger, Swinnerton-Dyer, Watson, Levenshtein, Odlyzko, Sloane and Musin. In this paper, we introduce and study a further generalization of the kissing numbers for convex bodies and obtain some exact results, in particular for balls in dimensions three, four and eight. References | Related Articles | Metrics

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NOTE ON PROJECTION BODIES OF ZONOTOPES WITH ${n+1}$ GENERATORS Martin Henk Acta mathematica scientia,Series B. 2025, 45 (1):  96-103.  DOI: 10.1007/s10473-025-0107-9 Abstract ( 85 )   Save We show that the volume of the projection body $\Pi(Z)$ of an $n$-dimensional zonotope $Z$ with $n+1$ generators and of volume $1$ is always exactly $2^n$. Moroever, we point out that an upper bound on the volume of $\Pi(K)$ of a centrally symmetric $n$-dimensional convex body of volume $1$ is at least $2^n (9/8)^{\lfloor n/3\rfloor}$. References | Related Articles | Metrics

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MODIFIED BRASCAMP-LIEB INEQUALITIES AND LOG-SOBOLEV INEQUALITIES FOR ONE-DIMENSIONAL LOG-CONCAVE MEASURE Denghui Wu, Jiazu Zhou Acta mathematica scientia,Series B. 2025, 45 (1):  104-117.  DOI: 10.1007/s10473-025-0108-8 Abstract ( 74 )   Save In this paper, we develop Maurey's and Bobkov-Ledoux's methods to prove modified Brascamp-Lieb inequalities and log-Sobolev inequalities for one-dimensional log-concave measure. To prove these inequalities, the harmonic Prékopa-Leindler inequality is used. We prove that these new inequalities are more efficient in estimating the variance and entropy for some functions with exponential terms. References | Related Articles | Metrics

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THE CONVEX SETS OF CONSTANT WIDTH CONSTRUCTED FROM OPPOSITE SECTORS Fengfan XIE, Yong YANG Acta mathematica scientia,Series B. 2025, 45 (1):  118-125.  DOI: 10.1007/s10473-025-0109-7 Abstract ( 66 )   Save This paper presents a method for constructing a convex set of constant width from opposite sectors. References | Related Articles | Metrics

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RECONSTRUCTION PROBLEMS OF CONVEX BODIES FROM EVEN $ {L_p}$ SURFACE AREA MEASURES Juewei Hu, Gangsong Leng Acta mathematica scientia,Series B. 2025, 45 (1):  126-142.  DOI: 10.1007/s10473-025-0110-1 Abstract ( 67 )   Save We build a computer program to reconstruct convex bodies using even $L_p$ surface area measures for $p\geq 1.$ Firstly, we transform the minimization problem $\mathcal{P}_1$, which is equivalent to solving the even $L_p$ Minkowski problem, into a convex optimization problem $\mathcal{P}_4$ with a finite number of constraints. This transformation makes it suitable for computational resolution. Then, we prove that the approximate solutions obtained by solving the problem $\mathcal{P}_4$ converge to the theoretical solution when $N$ and $k$ are sufficiently large. Finally, based on the convex optimization problem $\mathcal{P}_4$, we provide an algorithm for reconstructing convex bodies from even $L_p$ surface area measures, and present several examples implemented using MATLAB. References | Related Articles | Metrics

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LOG-CONCAVITY OF THE FIRST DIRICHLET EIGENFUNCTION OF SOME ELLIPTIC DIFFERENTIAL OPERATORS AND CONVEXITY INEQUALITIES FOR THE RELEVANT EIGENVALUE Andrea Colesanti Acta mathematica scientia,Series B. 2025, 45 (1):  143-152.  DOI: 10.1007/s10473-025-0111-0 Abstract ( 68 )   Save Given an open bounded subset $\Omega$ of $\mathbb{R}^{n}$ we consider the eigenvalue problem $\begin{equation*} \left\{ \begin{array}{lll} \Delta u-\langle\nabla u,\nabla V\rangle=-\lambda_V u,\quad u>0\quad&\mbox{in $\Omega$,}\\ u=0\quad&\mbox{on $\partial\Omega$,} \end{array} \right. \end{equation*}$ where $V$ is a given function defined in $\Omega$ and $\lambda_V$ is the relevant eigenvalue. We determine sufficient conditions on $V$ such that if $\Omega$ is convex, the solution $u$ is log-concave. We also determine sufficient conditions ensuring that $\lambda_V$, as a function of the set $\Omega$, verifies a convexity inequality with respect to the Minkowski addition of sets. References | Related Articles | Metrics

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KINEMATIC AND CROFTON FORMULAS FOR LINEAR GROUPS Ralph Howard Acta mathematica scientia,Series B. 2025, 45 (1):  153-160.  DOI: 10.1007/s10473-025-0112-z Abstract ( 51 )   Save Let $G_q\mathbb{R}^{n})$ be the Grassmannian of all linear $q$ dimensional subspaces of $\mathbb{R}^{n}$ and $I$ an integral invariant of $p+q-n$ dimensional submanifolds of $\mathbb{R}^{n}$. Then we give methods of evaluating Crofton type integral $ \int_{G_q(\mathbb{R}^{n})} I(M\cap L)\,{\rm d}L. $ The methods also work for various generalizations of $G_q(\mathbb{R}^{n})$ such as complex Grassmannians. References | Related Articles | Metrics

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THE CHORD GAUSS CURVATURE FLOW AND ITS $ {L_{p}}$ CHORD MINKOWSKI PROBLEM Jinrong Hu, Yong Huang, Jian Lu, Sinan Wang Acta mathematica scientia,Series B. 2025, 45 (1):  161-179.  DOI: 10.1007/s10473-025-0113-y Abstract ( 63 )   Save In this paper, the $L_{p}$ chord Minkowski problem is concerned. Based on the results shown in [20], we obtain a new existence result of solutions to this problem in terms of smooth measures by using a nonlocal Gauss curvature flow for $p>-n$ with $p\neq 0$. References | Related Articles | Metrics

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ON LEGENDRE AND LYZ ELLIPSOIDS Xinbao Lu, Ge Xiong, Jiangyan Tao Acta mathematica scientia,Series B. 2025, 45 (1):  180-188.  DOI: 10.1007/s10473-025-0114-x Abstract ( 60 )   Save We study the Legendre ellipsoid and the LYZ ellipsoid. First, we give a direct proof of the Cramer-Rao inequality for convex bodies. Second, we prove that origin-centered ellipsoids are the only convex bodies with identical Löwner and Legendre ellipsoids. Finally, we establish a mean width inequality for convex bodies whose LYZ ellipsoids are the Euclidean unit ball. References | Related Articles | Metrics

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ISOPERIMETRIC INEQUALITIES FOR INTEGRAL GEOMETRIC INVARIANTS OF RANDOM LINES Gaoyong Zhang Acta mathematica scientia,Series B. 2025, 45 (1):  189-199.  DOI: 10.1007/s10473-025-0115-9 Abstract ( 71 )   Save Isoperimetric type inequalities for integral geometric invariants of random lines in the Euclidean space are shown. Entropy inequalities of probability densities on the affine Grassmann manifold of lines are given. References | Related Articles | Metrics

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ON THE GAUSSIAN KINEMATIC FORMULA OF R. ADLER AND J. TAYLOR Joseph H.G. Fu Acta mathematica scientia,Series B. 2025, 45 (1):  200-214.  DOI: 10.1007/s10473-025-0116-8 Abstract ( 62 )   Save We apply methods of algebraic integral geometry to prove a special case of the Gaussian kinematic formula of Adler-Taylor. The idea, suggested already by Adler and Taylor, is to view the GKF as the limit of spherical kinematic formulas for spheres of large dimension $N$ and curvature $\frac 1 N$. References | Related Articles | Metrics

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STABILITY OF THE ISOPERIMETRIC INEQUALITY IN HYPERBOLIC PLANE Haizhong Li, Yao Wan Acta mathematica scientia,Series B. 2025, 45 (1):  215-227.  DOI: 10.1007/s10473-025-0117-7 Abstract ( 77 )   Save In this paper, we establish a stability estimate for the isoperimetric inequality of horospherically convex domains in hyperbolic plane. This estimate involves a relationship between the Hausdorff distance to a geodesic ball and the deficit in the isoperimetric inequality, where the coefficient of the deficit is a universal constant. References | Related Articles | Metrics

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A SURVEY ON THE ISOPERIMETRIC PROBLEM IN RIEMANNIAN MANIFOLDS Jiayu Li, Shujing Pan Acta mathematica scientia,Series B. 2025, 45 (1):  228-236.  DOI: 10.1007/s10473-025-0118-6 Abstract ( 85 )   Save This is a survey of the results in [14] regarding the isoperimetric problem in the Riemannian manifold. We consider a mean curvature type flow in the Riemannian manifold endowed with a non-trivial conformal vector field, which was firstly introduced by Guan and Li [8] in space forms. This flow preserves the volume of the bounded domain enclosed by a star-shaped hypersurface and decreases the area of hypersurface under certain conditions. We will prove the long time existence and convergence of the flow. As a result, the isoperimetric inequality for such a domain is established. References | Related Articles | Metrics

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ARCHIMEDES' PRINCIPLE OF FLOTATION AND FLOATING BODIES: CONSTRUCTION, EXTENSIONS AND RELATED PROBLEMS Chunyan Liu, Elisabeth M. Werner, Deping Ye, Ning Zhang Acta mathematica scientia,Series B. 2025, 45 (1):  237-256.  DOI: 10.1007/s10473-025-0119-5 Abstract ( 59 )   Save In this article, we explain how the famous Archimedes' principle of flotation can be used to construct various floating bodies. We survey some of the most important results regarding the floating bodies, including their relations with affine surface area and projection body, their extensions in different settings such as space forms and log-concave functions, and mention some associated open problems. References | Related Articles | Metrics

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ECHOS OF THE STEINER-LEHMUS EQUAL BISECTORS THEOREM Christoph Börgers, Eric L. Grinberg, Mehmet Orhon, Junhao Shen Acta mathematica scientia,Series B. 2025, 45 (1):  257-263.  DOI: 10.1007/s10473-025-0120-z Abstract ( 79 )   Save The Steiner-Lehmus equal bisectors theorem originated in the mid 19th century. Despite its age, it would have been accessible to Euclid and his contemporaries. The theorem remains evergreen, with new proofs continuing to appear steadily. The theorem has fostered discussion about the nature of proof itself, direct and indirect. Here we continue the momentum by providing a trigonometric proof, relatively short, based on an analytic estimate that leverages algebraic trigonometric identities. Many proofs of the theorem exist in the literature. Some of these contain key ideas that already appeared in C.L. Lehmus' 1850 proofs, not always with citation. In the aim of increasing awareness of and making more accessible Lehmus' proofs, we provide an annotated translation. We conclude with remarks on different proofs and relations among them. References | Related Articles | Metrics

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JERISON-LEE IDENTITIES AND SEMI-LINEAR SUBELLIPTIC EQUATIONS ON HEISENBERG GROUP Xinan Ma, Qianzhong Ou, Tian Wu Acta mathematica scientia,Series B. 2025, 45 (1):  264-279.  DOI: 10.1007/s10473-025-0121-y Abstract ( 86 )   Save In the study of the extremal for Sobolev inequality on the Heisenberg group and the Cauchy-Riemann(CR) Yamabe problem, Jerison-Lee found a three-dimensional family of differential identities for critical exponent subelliptic equation on Heisenberg group $\mathbb H^n$ by using the computer in [5]. They wanted to know whether there is a theoretical framework that would predict the existence and the structure of such formulae. With the help of dimension conservation and invariant tensors, we can answer the above question. References | Related Articles | Metrics

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CHARACTERIZATIONS OF BALLS AND ELLIPSOIDS BY INFINITESIMAL HOMOTHETIC CONDITIONS M. Angeles Alfonseca, Dmitry Ryabogin, Alina Stancu, Vladyslav Yaskin Acta mathematica scientia,Series B. 2025, 45 (1):  280-290.  DOI: 10.1007/s10473-025-0122-x Abstract ( 87 )   Save We prove that for a smooth convex body $K \subset \mathbb{R}^d, d\geq 2,$ with positive Gauss curvature, its homothety with a certain associated convex body implies that $K$ is either a ball or an ellipsoid, depending on the associated body considered. References | Related Articles | Metrics