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

Mathematika ◽  
2020 ◽  
Vol 66 (4) ◽  
pp. 1003-1027 ◽  
Author(s):  
Gennadiy Averkov ◽  
Christopher Borger ◽  
Ivan Soprunov
Keyword(s):  
Convex Bodies ◽  
Minkowski Sum ◽  
Mixed Volumes

scholarly journals Bézout-Type Inequality in Convex Geometry

2017 ◽  
Vol 2019 (16) ◽  
pp. 4950-4965 ◽  
Author(s):  
Jian Xiao
Keyword(s):  
Complex Geometry ◽  
Convex Geometry ◽  
Convex Bodies ◽  
Type Inequality ◽  
Mixed Volumes

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.

Determination of Boolean models by densities of mixed volumes

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.

scholarly journals REMARKS ABOUT MIXED DISCRIMINANTS AND VOLUMES

2014 ◽  
Vol 16 (02) ◽  
pp. 1350031 ◽  
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.

scholarly journals Flag representations of mixed volumes and mixed functionals of convex bodies

2018 ◽  
Vol 460 (2) ◽  
pp. 745-776 ◽  
Author(s):  
Daniel Hug ◽  
Jan Rataj ◽  
Wolfgang Weil
Keyword(s):  
Convex Bodies ◽  
Mixed Volumes

Applications of convex geometry to Minkowski sums of m ellipsoids in ℝN: Closed-form parametric equations and volume bounds

2021 ◽  
pp. 2140009
Author(s):  
Gregory S. Chirikjian ◽  
Bernard Shiffman
Keyword(s):  
Closed Form ◽  
Convex Geometry ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Dimensional Euclidean Space ◽  
Minkowski Sum ◽  
Principal Curvatures ◽  
Reverse Isoperimetric Inequality ◽  
Lower Volume ◽  
Minkowski Sums

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of [Formula: see text] arbitrary ellipsoids in [Formula: see text]-dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn–Minkowski inequality. A reverse isoperimetric inequality for convex bodies is also given.

scholarly journals Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
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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