Pinning a Line by Balls or Ovaloids in R
نویسندگان
چکیده
We show that if a line ` is an isolated line transversal to a finite familyF of (possibly intersecting) balls inR and no two balls are externally tangent on ` then there is a subset G ⊆ F of size at most 12 such that ` is an isolated line transversal to G. We generalize this result to families of semi-algebraic ovaloids.
منابع مشابه
Pinning a Line by Balls or Ovaloids in ℝ3
We show that if a line ` is an isolated line transversal to a finite familyF of (possibly intersecting) balls in R and no two balls are externally tangent on ` then there is a subfamily G ⊆ F of size at most 12 such that ` is an isolated line transversal to G. We generalize this result to families of semi-algebraic ovaloids.
متن کاملLower Bounds for Pinning Lines by Balls
A line l is a transversal to a family F of convex objects in R if it intersects every member of F . In this paper we show that for every integer d ≥ 3 there exists a family of 2d− 1 pairwise disjoint unit balls in R with the property that every subfamily of size 2d − 2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957. Key-w...
متن کاملOptimizing the Characteristics of the Motion of Steel Balls and their Impact on Shell Liners in SAG Mills
The equations governing the motion of steel balls and their impact onto shell liners in industrial Semi-Autogenous Grinding (SAG) mills are derived in full details by the authors and are used in order to determine the effective design variables for optimizing the working conditions of the mill and to avoid severe impacts which lead to the breakage of SAG mill shell liners. These design vari...
متن کاملWEAR RATE PREDICTION OF GRINDING MEDIA USING BPNN AND MLR MODELS IN GRINDING OF SULPHIDE ORES
Nowadays steel balls wear is a major problem in mineral processing industries and forms a significant part of the grinding cost. Different factors are effective on balls wear. It is needed to find models which are capable to estimate wear rate from these factors. In this paper a back propagation neural network (BPNN) and multiple linear regression (MLR) method have been used to predict wear rat...
متن کاملHadwiger and Helly-type theorems for disjoint unit spheres
We prove Helly-type theorems for line transversals to disjoint unit balls in R. In particular, we show that a family of n > 2d disjoint unit balls in R has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n > 4d − 1 disjoint unit balls in R admits a line transversal if any subfamily of ...
متن کامل