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On the hardness of minkowski addition and related operations

Hans Raj Tiwary
2007 Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07  
Since the convex hull of the union and the intersection of two polytopes relate naturally to the Minkowski sum via the Cayley trick and polarity, similar hardness results follow for these operations as  ...  For polytopes P, Q ⊂ R d we consider the intersection P ∩ Q; the convex hull of the union CH(P ∪ Q); and the Minkowski sum P + Q.  ...  HARDNESS OF MINKOWSKI ADDITION In We prove this by providing a polynomial-time Turing reduction from a coNP-Hard problem to this problem.  ... 
doi:10.1145/1247069.1247124 dblp:conf/compgeom/Tiwary07 fatcat:f6x4sofafbazvilofhf73rj5lm

On the Hardness of Computing Intersection, Union and Minkowski Sum of Polytopes

Hans Raj Tiwary
2008 Discrete & Computational Geometry  
For the intersection, we prove that computing the facets or the vertices of the intersection of two polytopes is NP-hard if one of them is given by vertices and the other by facets.  ...  For the Minkowski sum, we prove that enumerating the facets of P 1 + P 2 is NP-hard if P 1 and P 2 are specified by facets, or if P 1 is specified by vertices and P 2 is a polyhedral cone specified by  ...  The author would also like to thank Günter Rote for helpful discussions relating to Sect. 3.1 and Raimund Seidel for extremely helpful comments regarding the content and the structure of this paper and  ... 
doi:10.1007/s00454-008-9097-3 fatcat:7nrq464d3ffsrgp5y7l36ludbu

Computational complexity of determining which statements about causality hold in different space–time models

Vladik Kreinovich, Olga Kosheleva
2008 Theoretical Computer Science  
Causality is one of the most fundamental notions of physics. It is therefore important to be able to decide which statements about causality are correct in different models of space-time.  ...  In particular, we show that: • for Minkowski space-time, the decision problem is as difficult as Tarski's decision problem for elementary geometry, while • for a natural model of primordial space-time,  ...  To get a reduction, we need to describe: • real numbers, • the relationon real numbers, and • two arithmetic operations with these real numbers: . addition and . multiplication.  ... 
doi:10.1016/j.tcs.2008.06.028 fatcat:lqj44xo5ubgwhpoayynb3shvmy

Cell shape analysis of random tessellations based on Minkowski tensors [article]

Michael A. Klatt, Günter Last, Klaus Mecke, Claudia Redenbach, Fabian M. Schaller, Gerd E. Schröder-Turk
2016 arXiv   pre-print
We focus on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowski  ...  permanental point processes, and Gibbs hard-core and random sequential absorption processes as well as for Laguerre tessellations of polydisperse spheres and STIT- and Poisson hyperplane tessellations  ...  Acknowledgements We thank Andy Kraynik, Markus Spanner, and Richard Schielein for their data of the monodisperse foam, the equilibrium hard-sphere liquids, and the crystalline sphere packings, respectively  ... 
arXiv:1606.07653v1 fatcat:2qeqrmip2reelndrhz6ziqu7iq

Polygonal Boundary Evaluation of Minkowski Sums and Swept Volumes

Marcel Campen, Leif Kobbelt
2010 Computer graphics forum (Print)  
These sweep operations include Minkowski addition, offsetting, and sweeping along a discrete rigid motion trajectory.  ...  Many previous methods focus on the construction of a polygonal superset (containing self-intersections and spurious internal geometry) of the boundary of the volumes which are swept.  ...  The case of translationally sweeping a solid by another one is known as Minkowski addition, the special case of one of them being a sphere is referred to as offsetting.  ... 
doi:10.1111/j.1467-8659.2010.01770.x fatcat:hsc57ijjbbdkzlbr4siwe52h44

Even Minkowski valuations

Franz E. Schuster, Thomas Wannerer
2015 American Journal of Mathematics  
Moreover, the action of Alesker's Hard Lefschetz operators on these Minkowski valuations is explored in detail.  ...  Explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived.  ...  Acknowledgments The work of the authors was supported by the Austrian Science Fund (FWF), Project number: P 22388-N13.  ... 
doi:10.1353/ajm.2015.0041 fatcat:4clsbygegfdd7mnbvhhzzybsbu

Book Review: Foundations of special relativity: Kinematic axioms for Minkowski space-time

Patrick Suppes
1976 Bulletin of the American Mathematical Society  
and more intuitively characterizable operations or relations.  ...  In addition to the set of space-time events, he used particles, an ordering relation of beforeness on events, and, perhaps most importantly, a one-one signal-mapping from one particle onto another.  ...  Their sum is the total energy operator C, and to do quantum mechanics one must compute functions of it.  ... 
doi:10.1090/s0002-9904-1976-14044-x fatcat:wkwiozr4hrbqbc5rffxaev2vmq

Gravitational cusp anomalous dimension from AdS space

D. J. Miller, C. D. White
2012 Physical Review D  
The results may be useful in examining the relations between gauge and gravity theories.  ...  By making a particular coordinate transform, the calculation of the cusp anomalous dimension in QED or QCD can be related to the energy of a pair of static charges in Euclidean Anti-de-Sitter (AdS) space  ...  In addition, the cancellation of collinear singularities was generalised from soft to hard collinear configurations.  ... 
doi:10.1103/physrevd.85.104034 fatcat:o5ldgtyncfexpoch6m6vpymspm

Log-concavity properties of Minkowski valuations

Astrid Berg, Lukas Parapatits, Franz E. Schuster, Manuel Weberndorfer
2018 Transactions of the American Mathematical Society  
It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold.  ...  These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others.  ...  Acknowledgments The work of A. Berg, L. Parapatits, and F.E.  ... 
doi:10.1090/tran/7434 fatcat:jthuepbfdbf67aydkwwx4ythmu

Log-Concavity Properties of Minkowski Valuations [article]

Astrid Berg, Lukas Parapatits, Franz E. Schuster, Manuel Weberndorfer
2014 arXiv   pre-print
It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold.  ...  These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others.  ...  Acknowledgments The work of A. Berg, L. Parapatits, and F.E.  ... 
arXiv:1411.7891v1 fatcat:7c66mtcxmjayrk6s2qe3n37m2q

Even Minkowski Valuations [article]

Franz E. Schuster, Thomas Wannerer
2014 arXiv   pre-print
Moreover, the action of Alesker's Hard Lefschetz operators on these Minkowski valuations is explored in detail.  ...  Explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived.  ...  Acknowledgments The work of the authors was supported by the Austrian Science Fund (FWF), Project number: P 22388-N13.  ... 
arXiv:1411.1934v1 fatcat:4hth3ltawngpba4mo6uqbgon7e

Mixed integrals and related inequalities [article]

Vitali Milman, Liran Rotem
2012 arXiv   pre-print
In this paper we define an addition operation on the class of quasi-concave functions.  ...  We extend various classic inequalities, such as the Brunn-Minkowski and the Alexandrov-Fenchel inequality, to the functional setting.  ...  The authors would like to thank Prof. R. Schneider for his detailed comments regarding the written text.  ... 
arXiv:1210.4346v1 fatcat:hth4oagjyfbv5oxakh2yffzfwm

Page 6951 of Mathematical Reviews Vol. , Issue 93m [page]

1993 Mathematical Reviews  
Most morphological operations on images are based on the Min- kowski addition and subtraction; the latter were initially defined for sets in Euclidean space [see H.  ...  A reduction operator preserves connectivity if the inclusion relation between the connected components of the original figure and those of the reduced figure, as well as the inclusion relation between  ... 

On morphological operators based on rank filters

Pierre Soille
2002 Pattern Recognition  
We also highlight the relations between these operators and show that many are similar if not equivalent. An extensive bibliography is provided.  ...  In this paper, starting from a brief historical overview of rank order "lters and erosions/dilations in digital image processing, rank order based morphological operators are surveyed.  ...  Acknowledgements The author wishes to thank the referees for their very stimulating comments and suggestions and Henk Heijmans for his advice to include a discussion about the concept of adjunction.  ... 
doi:10.1016/s0031-3203(01)00047-4 fatcat:qoz2r25735er5hpowwp7iw2tu4

Effective Framework of kappa-Minkowski Spacetime and Its Application [article]

Yan-Gang Miao
2009 arXiv   pre-print
This paper has been withdrawn by the author because it needs to be rewritten completely.  ...  Müller-Kirsten for helpful discussions and the DFG (Deutsche Forschungsgemeinschaft) for financial support to his academic visit to the University of Kaiserslautern.  ...  Acknowledgments The author would like to thank H.J.W.  ... 
arXiv:0903.4257v3 fatcat:554cnpoqtngwjmqoqos7x2mmza
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