Improved Lp-mixed volume inequality for convex bodies

AbstractLet M be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large λ the number of lattice points in λM is given by G(λM) = V(λM) + O(λd−1−ε(d)) for some positive ε(d). Here we give for general convex bodies the weaker estimatewhere SZd (M) denotes the lattice surface area of M. The term SZd is optimal for all convex bodies and o(λd−1) cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of M.Further we deal with families {Pλ} of convex bodies where the only condition is that the inradius tends to infinity. Here we havewhere the convex body K satisfies some simple condition, V(Pλ; K; 1) is some mixed volume and S(Pλ) is the surface area of Pλ.

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Functional Geominimal Surface Area and Its Related Affine Isoperimetric Inequality

Journal of Function Spaces ◽

10.1155/2020/3039598 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8

Author(s):

Niufa Fang ◽

Jin Yang

Keyword(s):

Surface Area ◽

Isoperimetric Inequality ◽

Convex Bodies ◽

Mixed Volume ◽

Concave Functions ◽

Variation Formula ◽

First Variation ◽

The First Variation ◽

Geominimal Surface Area ◽

Affine Isoperimetric Inequality

The first variation of the total mass of log-concave functions was studied by Colesanti and Fragalà, which includes the Lp mixed volume of convex bodies. Using Colesanti and Fragalà’s first variation formula, we define the geominimal surface area for log-concave functions, and its related affine isoperimetric inequality is also established.

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Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Journal of Function Spaces ◽

10.1155/2018/9752178 ◽

2018 ◽

Vol 2018 ◽

pp. 1-16 ◽

Cited By ~ 3

Author(s):

Chang-Jian Zhao

Keyword(s):

Orlicz Space ◽

Convex Bodies ◽

Minkowski Inequality ◽

Mixed Volume ◽

Mixed Volumes ◽

Geometric Quantity ◽

First Order ◽

Special Cases ◽

Order Variation

Our main aim is to generalize the classical mixed volumeV(K1,…,Kn)and Aleksandrov-Fenchel inequality to the Orlicz space. In the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the Orlicz first-order variation of the mixed volume and call itOrlicz multiple mixed volumeof convex bodiesK1,…,Kn, andLn, denoted byVφ(K1,…,Kn,Ln), which involves(n+1)convex bodies inRn. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities ofLp-multiple mixed volumeVp(K1,…,Kn,Ln)are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yieldsLp-Aleksandrov-Fenchel inequality, Orlicz-Minkowski inequality, and Orlicz isoperimetric type inequalities. As application, a new Orlicz-Brunn-Minkowski inequality for Orlicz harmonic addition is established, which implies Orlicz-Brunn-Minkowski inequalities for the volumes and quermassintegrals.

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Monotonicity and concavity of integral functionals involving area measures of convex bodies

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500334 ◽

2017 ◽

Vol 19 (02) ◽

pp. 1650033 ◽

Cited By ~ 1

Author(s):

Andrea Colesanti ◽

Daniel Hug ◽

Eugenia Saorín Gómez

Keyword(s):

Necessary Conditions ◽

Broad Class ◽

Convex Bodies ◽

Sufficient Conditions ◽

Type Inequality ◽

Mixed Volume ◽

Necessary And Sufficient Conditions ◽

Integral Functionals ◽

Necessary And Sufficient

For a broad class of integral functionals defined on the space of [Formula: see text]-dimensional convex bodies, we establish necessary and sufficient conditions for monotonicity, and necessary conditions for the validity of a Brunn–Minkowski type inequality. In particular, we prove that a Brunn–Minkowski type inequality implies monotonicity, and that a general Brunn–Minkowski type inequality is equivalent to the functional being a mixed volume.

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Convex Bodies: The Brunn–Minkowski Theory

10.1017/cbo9780511526282 ◽

1993 ◽

Cited By ~ 891

Author(s):

Rolf Schneider

Keyword(s):

Convex Bodies

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Excess entropy of the systems of molecules differing in size and shape

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19830192 ◽

1983 ◽

Vol 48 (1) ◽

pp. 192-198 ◽

Cited By ~ 5

Author(s):

Tomáš Boublík

Keyword(s):

Equation Of State ◽

Hard Spheres ◽

Excess Entropy ◽

Convex Bodies ◽

Pure Substance ◽

Liquid Equilibrium ◽

Simple Rules ◽

Different Shapes ◽

The Mean

The excess entropy of mixing of mixtures of hard spheres and spherocylinders is determined from an equation of state of hard convex bodies. The obtained dependence of excess entropy on composition was used to find the accuracy of determining ΔSE from relations employed for the correlation and prediction of vapour-liquid equilibrium. Simple rules were proposed for establishing the mean parameter of nonsphericity for mixtures of hard bodies of different shapes allowing to describe the P-V-T behaviour of solutions in terms of the equation of state fo pure substance. The determination of ΔSE by means of these rules is discussed.

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Some Hermite–Hadamard type integral inequalities for convex functions defined on convex bodies in ℝn\mathbb{R}^{n}

Journal of Applied Analysis ◽

10.1515/jaa-2020-2005 ◽

2020 ◽

Vol 26 (1) ◽

pp. 67-77 ◽

Cited By ~ 1

Author(s):

Silvestru Sever Dragomir

Keyword(s):

Euclidean Space ◽

Convex Functions ◽

Convex Bodies ◽

Integral Inequalities ◽

Divergence Theorem ◽

Several Variables ◽

Type Integral ◽

Bounded Convex

AbstractIn this paper, by the use of the divergence theorem, we establish some integral inequalities of Hermite–Hadamard type for convex functions of several variables defined on closed and bounded convex bodies in the Euclidean space {\mathbb{R}^{n}} for any {n\geq 2}.

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