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CROOKED HALFSPACES
, 2014
"... Dedicated to the memory of Robert Miner Abstract. We develop the Lorentzian geometry of a crooked halfspace in 2 + 1dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving an e ..."
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Dedicated to the memory of Robert Miner Abstract. We develop the Lorentzian geometry of a crooked halfspace in 2 + 1dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving
Infeasibility of Systems of Halfspaces
, 2001
"... An oriented hyperplane is a hyperplane with designated good and bad sides. The infeasibility of a cell in an arrangement ~ A of oriented hyperplanes is the number of hyperplanes with this cell on the bad side. Wit MinInf( ~ A) we denote the minimum infeasibility of a cell in the arrangement. A s ..."
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Cited by 1 (0 self)
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subset of hyperplanes of ~ A is called an infeasible subsystem if every cell in the induced subarrangement has positive infeasibility. With MaxDis( ~ A) we denote the maximal number of disjoint infeasible subsystems of ~ A. For every arrangement ~ A of oriented hyperplanes MinInf( ~ A) Max
εNets for Halfspaces Revisited∗
, 2014
"... “It is a damn poor mind indeed which can’t think of at least two ways to spell any word.” – Andrew Jackson Given a set P of n points in R3, we show that, for any ε> 0, there exists an εnet of P for halfspace ranges, of size O(1/ε). We give five proofs of this result, which are arguably simpler t ..."
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“It is a damn poor mind indeed which can’t think of at least two ways to spell any word.” – Andrew Jackson Given a set P of n points in R3, we show that, for any ε> 0, there exists an εnet of P for halfspace ranges, of size O(1/ε). We give five proofs of this result, which are arguably simpler
Tropical Convexity, Halfspace Arrangements and Optimization
, 2008
"... This master thesis investigates discrete geometry in the tropical semiring (R,min,+), setting its main focus on convex polytopes and halfspace arrangements. Specifically, tropical analogs to results in classical discrete geometry are presented. Results include tropical versions of the separation of ..."
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This master thesis investigates discrete geometry in the tropical semiring (R,min,+), setting its main focus on convex polytopes and halfspace arrangements. Specifically, tropical analogs to results in classical discrete geometry are presented. Results include tropical versions of the separation
Entropy geometry and disjointness for zerodimensional algebraic actions
 J. Reine Angew. Math
, 2005
"... Abstract. We show that many algebraic actions of higherrank abelian groups on zerodimensional groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov–Rokhlin formula for halfspace entropies. We discuss some mutual disjoint ..."
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Cited by 7 (2 self)
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Abstract. We show that many algebraic actions of higherrank abelian groups on zerodimensional groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov–Rokhlin formula for halfspace entropies. We discuss some mutual
Adelic amoebas disjoint from open halfpsaces
, 2007
"... Abstract. We show that a conjecture of Einsiedler, Kapranov, and Lind on adelic amoebas of subvarieties of tori and their intersections with open halfspaces of complementary dimension is false for subvarieties of codimension greater than one that have degenerate projections to smaller dimensional to ..."
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Cited by 3 (2 self)
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Abstract. We show that a conjecture of Einsiedler, Kapranov, and Lind on adelic amoebas of subvarieties of tori and their intersections with open halfspaces of complementary dimension is false for subvarieties of codimension greater than one that have degenerate projections to smaller dimensional
Binary Plane Partitions for Disjoint Line Segments
"... A binary space partition (BSP) for a set of disjoint objects in Euclidean space is a recursive decomposition, where each step partitions the space (and some of the objects) along a hyperplane and recurses on the objects clipped in each of the two open halfspaces. The size of a BSP is defined as the ..."
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Cited by 2 (1 self)
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A binary space partition (BSP) for a set of disjoint objects in Euclidean space is a recursive decomposition, where each step partitions the space (and some of the objects) along a hyperplane and recurses on the objects clipped in each of the two open halfspaces. The size of a BSP is defined
Nonzero boundaries of Leibniz halfspaces∗
, 2003
"... It is proved that for any d ≥ 3, there exists a norm ‖ · ‖ and two points a, b in Rd, such that the boundary of the Leibniz halfspace H(a, b) = {x ∈ Rd: ‖x − a ‖ ≤ ‖x − b‖} has nonzero Lebesgue measure. When d = 2, it is known that the boundary must have zero Lebesgue measure. ..."
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It is proved that for any d ≥ 3, there exists a norm ‖ · ‖ and two points a, b in Rd, such that the boundary of the Leibniz halfspace H(a, b) = {x ∈ Rd: ‖x − a ‖ ≤ ‖x − b‖} has nonzero Lebesgue measure. When d = 2, it is known that the boundary must have zero Lebesgue measure.
Results 1  10
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1,899