Flag representations of mixed volumes and mixed functionals of convex bodies

Abstract Inspired by a result of Soprunov and Zvavitch, we present a Bézout type inequality for mixed volumes, which holds true for any convex bodies and improves the previous result. The key ingredient is the reverse Khovanskii–Teissier inequality for convex bodies, which was obtained in our previous work and inspired by its correspondence in complex geometry.

Download Full-text

Determination of Boolean models by densities of mixed volumes

Advances in Applied Probability ◽

10.1017/apr.2019.5 ◽

2019 ◽

Vol 51 (01) ◽

pp. 116-135

Author(s):

Daniel Hug ◽

Wolfgang Weil

Keyword(s):

Uniqueness Theorem ◽

Representation Theorem ◽

Convex Bodies ◽

Boolean Models ◽

Mixed Volumes ◽

Shape Information ◽

Translation Invariant ◽

Mean Values ◽

Particle Process

AbstractIn Weil (2001) formulae were proved for stationary Boolean models Z in ℝd with convex or polyconvex grains, which express the densities (specific mean values) of mixed volumes of Z in terms of related mean values of the underlying Poisson particle process X. These formulae were then used to show that in dimensions 2 and 3 the densities of mixed volumes of Z determine the intensity γ of X. For d = 4, a corresponding result was also stated, but the proof given was incomplete, since in the formula for the density of the Euler characteristic V̅0(Z) of Z a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed out in Goodey and Weil (2002), where it was also explained that a new decomposition result for mixed volumes and mixed translative functionals would be needed to complete the proof. Such a general decomposition result has recently been proved by Hug, Rataj, and Weil (2013), (2018) and is based on flag measures of the convex bodies involved. Here, we show that such flag representations not only lead to a correct derivation of the four-dimensional result, but even yield a corresponding uniqueness theorem in all dimensions. In the proof of the latter we make use of Alesker’s representation theorem for translation invariant valuations. We also discuss which shape information can be obtained in this way and comment on the situation in the nonstationary case.

Download Full-text

A product integral representation of mixed volumes of two convex bodies

Advances in Geometry ◽

10.1515/advgeom-2012-0044 ◽

2013 ◽

Vol 13 (4) ◽

Cited By ~ 3

Author(s):

Daniel Hug ◽

Jan Rataj ◽

Wolfgang Weil

Keyword(s):

Integral Representation ◽

Convex Bodies ◽

Mixed Volumes ◽

Product Integral

Download Full-text

REMARKS ABOUT MIXED DISCRIMINANTS AND VOLUMES

Communications in Contemporary Mathematics ◽

10.1142/s0219199713500314 ◽

2014 ◽

Vol 16 (02) ◽

pp. 1350031 ◽

Cited By ~ 4

Author(s):

S. ARTSTEIN-AVIDAN ◽

D. FLORENTIN ◽

Y. OSTROVER

Keyword(s):

Fisher Information ◽

Convex Sets ◽

Compact Convex ◽

Convex Bodies ◽

Mixed Volumes ◽

Information Functional ◽

Geometric Analogue

In this note we prove certain inequalities for mixed discriminants of positive semi-definite matrices, and mixed volumes of compact convex sets in ℝn. Moreover, we discuss how the latter are related to the monotonicity of an information functional on the class of convex bodies, which is a geometric analogue of the classical Fisher information.

Download Full-text

INEQUALITIES BETWEEN MIXED VOLUMES OF CONVEX BODIES: VOLUME BOUNDS FOR THE MINKOWSKI SUM

Mathematika ◽

10.1112/mtk.12055 ◽

2020 ◽

Vol 66 (4) ◽

pp. 1003-1027 ◽

Cited By ~ 1

Author(s):

Gennadiy Averkov ◽

Christopher Borger ◽

Ivan Soprunov

Keyword(s):

Convex Bodies ◽

Minkowski Sum ◽

Mixed Volumes

Download Full-text

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.

Download Full-text