Minkowski Inequality Research Articles

Mixed volume is an important notion in convex geometry, which is the extension of volume and surface area. The Minkowski inequality for mixed volume plays a vital role in convex geometry. This paper obtains that mixed volume under Steiner symmetrization is monotonic and decreasing, and a concise proof of the general Minkowski inequality by Steiner symmetrization is obtained.

<abstract><p>In this paper, we introduced the concept of $ C $-star bodies in a fixed pointed closed convex cone $ C $ and studied the dual mixed volume for $ C $-star bodies. For $ C $-star bodies, we established the corresponding dual Brunn-Minkowski inequality, dual Minkowski inequality, and dual Aleksandrov-Fenchel inequality. Moveover, we found that the dual Brunn-Minkowski inequality for $ C $-star bodies can strengthen the Brunn-Minkowski inequality for $ C $-coconvex sets.</p></abstract>

In the paper, our main aim is to introduce a new ?-mixed affine surface area ??,p(K, L) of convex bodies, which obeys classical basic properties. The new affine geometric quantity in special case yields the classical Lp-affine surface area ?p(K), Lp-mixed affine surface area ?p(K, L) and the newly established Lpq-mixed affine surface area ?p,q(K, L), respectively. As an application, we establish a ?-Minkowski inequality for the ?-mixed affine surface area, which follows the classical Minkowski inequality for mixed affine surface area ??1(K, L), Lp-Minkowski inequality for Lp-affine surface area and Lpq-Minkowski inequality for Lpq-mixed affine surface area, respectively.

In this paper, we explore questions regarding the Minkowski sum of the boundaries of convex sets. Motivated by a question suggested to us by V. Milman regarding the volume of [Formula: see text] where [Formula: see text] and [Formula: see text] are convex bodies, we prove sharp volumetric lower bounds for the Minkowski average of the boundaries of sets with connected boundary, as well as some related results.

We consider the problem of finding the best function φn:[0,1]→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi _n:[0,1]\\rightarrow {\\mathbb {R}}$$\\end{document} such that for any pair of convex bodies K,L∈Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K,L\\in {\\mathbb {R}}^n$$\\end{document} the following Brunn–Minkowski type inequality holds |K+θL|1n≥φn(θ)(|K|1n+|L|1n),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} |K+_\ heta L|^\\frac{1}{n}\\ge \\varphi _n(\ heta )(|K|^\\frac{1}{n}+|L|^\\frac{1}{n}), \\end{aligned}$$\\end{document}where K+θL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K+_\ heta L$$\\end{document} is the θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta $$\\end{document}-convolution body of K and L. We prove a sharp inclusion of the family of Ball’s bodies of an α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-concave function in its super-level sets in order to provide the best possible function in the range 34n≤θ≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( \\frac{3}{4}\\right) ^n\\le \ heta \\le 1$$\\end{document}, characterizing the equality cases.

In this paper, we prove some inequalities for Riemann–Lebesgue integrable functions when the considered integration is obtained via a non-additive measure, including the reverse Hölder inequality and the reverse Minkowski inequality. Then, we generalize these inequalities to the framework of a multivalued case, in particular for Riemann–Lebesgue integrable interval-valued multifunctions, and obtain some inequalities, such as a Minkowski-type inequality, a Beckenbach-type inequality and some generalizations of Hölder inequalities.

Drawn-out dual Lp-Brunn-Minkowski inequality, some ordinary and useful characters and the uniqueness in the matter of Lp dual mixed affine quermassintegrals are discussed, and the equivalence of the inequality concerning Lp harmonic combinations, and its dual Lp- Minkowski inequality is proved.

Abstract In this work, it is assumed that the norm over bicomplex numbers is the hyperbolic ( 𝔻 {\mathbb{D}} -valued) norm. In this paper, we provide an overview of bicomplex Lebesgue spaces and investigate some of their geometric properties, including 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -convexity, 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -strict convexity, and 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -uniform convexity. Moreover, the basic inequalities such as 𝔻 {\mathbb{D}} -Hölder’s inequality and 𝔻 {\mathbb{D}} -Minkowski inequality for bicomplex Lebesgue spaces are presented, used to show geometric properties.

We present the generalizations of Hölder's inequality and Minkowski's inequality along with the generalizations of Aczel's, Popoviciu's, Lyapunov's and Bellman's inequalities. Some applications for the metric spaces, normed spaces, Banach spaces, sequence spaces and integral inequalities are further specified. It is shown that $({\mathbb{R}}^n,d)$ and $\left(l_p,d_{m,p}\right)$ are complete metric spaces and $({\mathbb{R}}^n,{\left\|x\right\|}_m)$ and $\left(l_p,{\left\|x\right\|}_{m,p}\right)$ are $\frac{1}{m}-$Banach spaces. Also, it is deduced that $\left(b^{r,s}_{p,1},{\left\|x\right\|}_{r,s,m}\right)$ is a $\frac{1}{m}-$normed space.

We prove a sharp stability result for the Brunn–Minkowski inequality for A,B\subset\mathbb{R}^2 . Assuming that the Brunn–Minkowski deficit \delta=|A+B|^{1/2}/(|A|^{1/2}+|B|^{1/2})-1 is sufficiently small in terms of t=|A|^{1/2}/(|A|^{1/2}+|B|^{1/2}) , there exist homothetic convex sets K_A \supset A and K_B\supset B such that \frac{|K_A\setminus A|}{|A|}+\frac{|K_B\setminus B|}{|B|} \le C t^{-{1/2}}\delta^{1/2} . The key ingredient is to show for every \epsilon,t>0 , if \delta is sufficiently small then |\!\operatorname{co}(A+B)\setminus (A+B)|\le (1+\epsilon)(|\!\operatorname{co}(A)\setminus A|+|\!\operatorname{co}(B)\setminus B|) .

We interpret the log-Brunn–Minkowski conjecture of Böröczky–Lutwak–Yang–Zhang as a spectral problem in centro-affine differential geometry. In particular, we show that the Hilbert–Brunn–Minkowski operator coincides with the centro-affine Laplacian, thus obtaining a new avenue for tackling the conjecture using insights from affine differential geometry. As every strongly convex hypersurface in \mathbb{R}^n is a centro-affine unit sphere, it has constant centro-affine Ricci curvature equal to n-2 , in stark contrast to the standard weighted Ricci curvature of the associated metric-measure space, which will in general be negative. In particular, we may use the classical argument of Lichnerowicz and a centro-affine Bochner formula to give a new proof of the Brunn–Minkowski inequality. For origin-symmetric convex bodies enjoying fairly generous curvature pinching bounds (improving with dimension), we are able to show global uniqueness in the L^p - and log-Minkowski problems, as well as the corresponding global L^p - and log-Minkowski conjectured inequalities. As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body \bar K in \mathbb{R}^n , there exists an origin-symmetric convex body K with \bar K \subset K \subset 8 \bar K such that K satisfies the log-Minkowski conjectured inequality, and such that K is uniquely determined by its cone-volume measure V_K . If \bar K is not extremely far from a Euclidean ball to begin with, an analogous isometric result, where 8 is replaced by 1+\varepsilon , is obtained as well.

This article considers a general family of weighted fractional integral operators and utilizes this general operator to establish numerous reverse Minkowski inequalities. When it comes to understanding and investigating convexity and inequality, symmetry is crucial. It provides insightful explanations, clearer explanations, and useful methods to help with the learning of key mathematical ideas. The kernel of the general family of weighted fractional integral operators is related to a wide variety of extensions and generalizations of the Mittag-Leffler function and the Hurwitz-Lerch zeta function. It delves into the applications of fractional-order integral and derivative operators in mathematical and engineering sciences. Furthermore, this article derives specific cases for selected functions and presents various applications to illustrate the obtained results. Additionally, novel applications involving the Digamma function are introduced.

Henstock and Macbeath asked in 1953 whether the Brunn–Minkowski inequality can be generalized to nonabelian locally compact groups; questions along the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. The proof follows an induction on dimension strategy; new ingredients include an understanding of the role played by maximal compact subgroups of Lie groups, a necessary modified form of the inequality which is also applicable to nonunimodular locally compact groups, and a proportionated averaging trick.

We establish fractional versions of generalizations of the Schweitzer, Kantorovich, Polya–Szego, Cassels, Greub–Rheinboldt, and reverse Minkowski inequalities on time scales. We present that fractional P´olya–Szego’s dynamic inequality generalizes Cassels’ inequality. Time scales calculus unifies and extends discrete, continuous, quantum versions of results.

Abstract In this paper, we obtain the functional Orlicz–Brunn–Minkowski inequality and the functional Orlicz–Minkowski inequality for q‐torsional rigidity in the smooth category. Furthermore, using an approximation method, we give the general functional Orlicz–Brunn–Minkowski inequality for q‐torsional rigidity. As a corollary, we reveal that the functional Orlicz–Brunn–Minkowski inequality is equivalent to the functional Orlicz–Minkowski inequality for q‐torsional rigidity in the smooth category. We also give some applications with respect to these two inequalities.

Abstract Using harmonic mean curvature flow, we establish a sharp Minkowski-type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard $3$-manifolds. This inequality also improves the known estimates for total mean curvature in hyperbolic $3$-space. As an application, we obtain a Bonnesen-style isoperimetric inequality for surfaces with convex distance function in nonpositively curved $3$-spaces, via monotonicity results for total mean curvature. This connection between the Minkowski and isoperimetric inequalities is extended to Cartan–Hadamard manifolds of any dimension.

Abstract We introduce a new operation between nonnegative integrable functions on , that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature of this operation is that the Lebesgue integral of the geometric combination equals the geometric mean of the two separate integrals; as a natural consequence, we derive a new functional inequality of Prékopa–Leindler type. When applied to the characteristic functions of two measurable sets, their geometric combination provides a set whose volume equals the geometric mean of the two separate volumes. In the framework of convex bodies, by comparing the geometric combination with the 0‐sum, we get an alternative proof of the log‐Brunn–Minkowski inequality for unconditional convex bodies and for convex bodies with n symmetries.

Abstract In the paper, our main aim is to generalize the q th dual volume to Lp space, and introduce pq th-dual mixed volume by calculating the first order variation of q th dual volumes. We establish the Lpq -Minkowski inequality for pq th-dual mixed volumes and Lpq -Brunn-Minkowski inequality for the q th-dual volumes, respectively. The new inequalities in special case yield some new dual inequalities for the q th-dual volumes.

We prove and discuss some new refined Hölder inequalities for any p>1 and also a reversed version for 0< p<1. The key is to use the concepts of superquadraticity, strong convexity, and to first prove the corresponding refinements of the Young and reversed Young inequalities. Refinements of the Minkowski and reversed Minkowski inequalities are also given.

In this article, we introduce a class of functions mathfrak{U}(mathfrak{p}) with integral representation defined over a measure space with σ-finite measure. The main purpose of this paper is to extend the Minkowski and related inequalities by considering general kernels. As a consequence of our general results, we connect our results with various variants for the fractional integrals operators. Such applications have wide use and importance in the field of applied sciences.