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Laurent Meersseman - One of the best experts on this subject based on the ideXlab platform.
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Real quadrics in Cn, complex manifolds and Convex Polytopes
Acta Mathematica, 2006Co-Authors: Frédéric Bosio, Laurent MeerssemanAbstract:In this paper, we investigate the topology of a class of non-Kahler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics C n which are invariant with respect to the natural action of the real torus (S 1) n onto C n . The quotient space is a simple Convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on Convex Polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple Convex polytope. Finally, as a nice consequence, we obtain that affine non-Kahler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kahler situation.
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Real quadrics in $\Bbb C^n$, complex manifolds and Convex Polytopes
Acta Mathematica, 2006Co-Authors: Frédéric Bosio, Laurent MeerssemanAbstract:In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics in $\Bbb C^n$ which are invariant with respect to the natural action of the real torus $(\Bbb S^1)^n$ onto $\Bbb C^n$. The quotient space is a simple Convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on Convex Polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple Convex polytope. Finally, as a nice consequence, we obtain that affine non Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.
Eleni Tzanaki - One of the best experts on this subject based on the ideXlab platform.
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The Maximum Number of Faces of the Minkowski Sum of Two Convex Polytopes
Discrete & Computational Geometry, 2016Co-Authors: Menelaos I. Karavelas, Eleni TzanakiAbstract:We derive tight bounds for the maximum number of k -faces, $$0\le k\le d-1$$ 0 ≤ k ≤ d - 1 , of the Minkowski sum, $$P_1+P_2$$ P 1 + P 2 , of two d -dimensional Convex Polytopes $$P_1$$ P 1 and $$P_2$$ P 2 , as a function of the number of vertices of the Polytopes. For even dimensions $$d\ge 2$$ d ≥ 2 , the maximum values are attained when $$P_1$$ P 1 and $$P_2$$ P 2 are cyclic d -Polytopes with disjoint vertex sets. For odd dimensions $$d\ge 3$$ d ≥ 3 , the maximum values are attained when $$P_1$$ P 1 and $$P_2$$ P 2 are $$\lfloor \frac{d}{2}\rfloor $$ ⌊ d 2 ⌋ -neighborly d -Polytopes, whose vertex sets are chosen appropriately from two distinct d -dimensional moment-like curves.
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Convex hulls of spheres and Convex hulls of disjoint Convex Polytopes
Computational Geometry, 2013Co-Authors: Menelaos I. Karavelas, Raimund Seidel, Eleni TzanakiAbstract:Given a set @S of spheres in E^d, with d>=3 and d odd, having a constant number of m distinct radii @r"1,@r"2,...,@r"m, we show that the worst-case combinatorial complexity of the Convex hull of @S is @Q(@?"1"= "j"= =3 odd, where n"i spheres have radius @r"i, i=1,2, and @r"2 @r"1, such that their Convex hull has combinatorial complexity @W(n"1n"2^@?^d^2^@?+n"2n"1^@?^d^2^@?). Our construction is then generalized to the case where the spheres have m>=3 distinct radii. For the upper bound, we reduce the sphere Convex hull problem to the problem of computing the worst-case combinatorial complexity of the Convex hull of a set of m disjoint d-dimensional Convex Polytopes in E^d^+^1, where d>=3 odd, a problem which is of independent interest. More precisely, we show that the worst-case combinatorial complexity of the Convex hull of a set of m disjoint d-dimensional Convex Polytopes in E^d^+^1 is O(@?"1"= "j"= =3. Finally, we discuss how to compute Convex hulls of spheres with a constant number of distinct radii, or Convex hulls of a constant number of disjoint Convex Polytopes.
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the maximum number of faces of the minkowski sum of two Convex Polytopes
Symposium on Discrete Algorithms, 2012Co-Authors: Menelaos I. Karavelas, Eleni TzanakiAbstract:We derive tight bounds for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P1 ⊕ P2, of two d-dimensional Convex Polytopes P1 and P2, as a function of the number of vertices of the Polytopes. For even dimensions d ≥ 2, the maximum values are attained when P1 and P2 are cyclic d-Polytopes with disjoint vertex sets. For odd dimensions d ≥ 3, the maximum values are attained when P1 and P2 are [d/2]-neighborly d-Polytopes, whose vertex sets are chosen appropriately from two distinct d-dimensional moment-like curves.
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Symposium on Computational Geometry - Convex hulls of spheres and Convex hulls of Convex Polytopes lying on parallel hyperplanes
Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG '11, 2011Co-Authors: Menelaos I. Karavelas, Eleni TzanakiAbstract:Given a set Σ of spheres in Ed, with d≥3 and d odd, having a fixed number of m distinct radii ρ1,ρ2,...,ρm, we show that the worst-case combinatorial complexity of the Convex hull CHd(Σ) of Σ is Θ(Σ{1≤i≠j≤m}ninj⌊ d/2 ⌋), where ni is the number of spheres in Σ with radius ρi. Our bound refines the worst-case upper and lower bounds on the worst-case combinatorial complexity of CHd(Σ) for all odd d≥3. To prove the lower bound, we construct a set of Θ(n1+n2) spheres in Ed, with d≥3 odd, where ni spheres have radius ρi, i=1,2, and ρ_2≠ρ1, such that their Convex hull has combinatorial complexity Ω(n1n2⌊ d/2 ⌋+n2n1⌊ d/2 ⌋). Our construction is then generalized to the case where the spheres have m≥3 distinct radii. For the upper bound, we reduce the sphere Convex hull problem to the problem of computing the worst-case combinatorial complexity of the Convex hull of a set of m d-dimensional Convex Polytopes lying on m parallel hyperplanes in Ed+1, where d≥3 odd, a problem which is of independent interest. More precisely, we show that the worst-case combinatorial complexity of the Convex hull of a set P{1,P2,...,Pm} of m d-dimensional Convex Polytopes lying on m parallel hyperplanes of Ed+1 is O(Σ1≤i≠j≤mninj⌊ d/2 ⌋), where ni is the number of vertices of Pi. This bound is an improvement over the worst-case bound on the combinatorial complexity of the Convex hull of a point set where we impose no restriction on the points' configuration; using the lower bound construction for the sphere Convex hull problem, it is also shown to be tight for all odd d≥3. Finally: (1) we briefly discuss how to compute Convex hulls of spheres with a fixed number of distinct radii, or Convex hulls of a fixed number of Polytopes lying on parallel hyperplanes; (2) we show how our tight bounds for the parallel polytope Convex hull problem, yield tight bounds on the combinatorial complexity of the Minkowski sum of two Convex Polytopes in Ed; and (3) we state some open problems and directions for future work.
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Convex hulls of spheres and Convex hulls of Convex Polytopes lying on parallel hyperplanes
arXiv: Computational Geometry, 2009Co-Authors: Menelaos I. Karavelas, Eleni TzanakiAbstract:Given a set $\Sigma$ of spheres in $\mathbb{E}^d$, with $d\ge{}3$ and $d$ odd, having a fixed number of $m$ distinct radii $\rho_1,\rho_2,...,\rho_m$, we show that the worst-case combinatorial complexity of the Convex hull $CH_d(\Sigma)$ of $\Sigma$ is $\Theta(\sum_{1\le{}i\ne{}j\le{}m}n_in_j^{\lfloor\frac{d}{2}\rfloor})$, where $n_i$ is the number of spheres in $\Sigma$ with radius $\rho_i$. To prove the lower bound, we construct a set of $\Theta(n_1+n_2)$ spheres in $\mathbb{E}^d$, with $d\ge{}3$ odd, where $n_i$ spheres have radius $\rho_i$, $i=1,2$, and $\rho_2\ne\rho_1$, such that their Convex hull has combinatorial complexity $\Omega(n_1n_2^{\lfloor\frac{d}{2}\rfloor}+n_2n_1^{\lfloor\frac{d}{2}\rfloor})$. Our construction is then generalized to the case where the spheres have $m\ge{}3$ distinct radii. For the upper bound, we reduce the sphere Convex hull problem to the problem of computing the worst-case combinatorial complexity of the Convex hull of a set of $m$ $d$-dimensional Convex Polytopes lying on $m$ parallel hyperplanes in $\mathbb{E}^{d+1}$, where $d\ge{}3$ odd, a problem which is of independent interest. More precisely, we show that the worst-case combinatorial complexity of the Convex hull of a set $\{\mathcal{P}_1,\mathcal{P}_2,...,\mathcal{P}_m\}$ of $m$ $d$-dimensional Convex Polytopes lying on $m$ parallel hyperplanes of $\mathbb{E}^{d+1}$ is $O(\sum_{1\le{}i\ne{}j\le{}m}n_in_j^{\lfloor\frac{d}{2}\rfloor})$, where $n_i$ is the number of vertices of $\mathcal{P}_i$. We end with algorithmic considerations, and we show how our tight bounds for the parallel polytope Convex hull problem, yield tight bounds on the combinatorial complexity of the Minkowski sum of two Convex Polytopes in $\mathbb{E}^d$.
Li Zhang - One of the best experts on this subject based on the ideXlab platform.
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the minimax risk of truncated series estimators for symmetric Convex Polytopes
International Symposium on Information Theory, 2012Co-Authors: Adel Javanmard, Li ZhangAbstract:We study the optimality of the minimax risk of truncated series estimators over symmetric Convex Polytopes. We show that the optimal truncated series estimator is within O(log m) factor of the optimal if the polytope is defined by m hyperplanes. This represents the first such bounds towards general Convex bodies. In proving our result, we first define a geometric quantity, called the approximation radius, for lower bounding the minimax risk. We then derive our bounds by establishing a connection between the approximation radius and the Kolmogorov width, the quantity that provides upper bounds for the truncated series estimator. Besides, our proof contains several ingredients which might be of independent interest: 1. The notion of approximation radius depends on the volume of the body. It is an intuitive notion and is flexible to yield strong minimax lower bounds; 2. The connection between the approximation radius and the Kolmogorov width is a consequence of a novel duality relationship on the Kolmogorov width, developed by utilizing some classical results from Convex geometry [1], [18], [6].
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the minimax risk of truncated series estimators for symmetric Convex Polytopes
arXiv: Statistics Theory, 2012Co-Authors: Adel Javanmard, Li ZhangAbstract:We study the optimality of the minimax risk of truncated series estimators for symmetric Convex Polytopes. We show that the optimal truncated series estimator is within $O(\log m)$ factor of the optimal if the polytope is defined by $m$ hyperplanes. This represents the first such bounds towards general Convex bodies. In proving our result, we first define a geometric quantity, called the \emph{approximation radius}, for lower bounding the minimax risk. We then derive our bounds by establishing a connection between the approximation radius and the Kolmogorov width, the quantity that provides upper bounds for the truncated series estimator. Besides, our proof contains several ingredients which might be of independent interest: 1. The notion of approximation radius depends on the volume of the body. It is an intuitive notion and is flexible to yield strong minimax lower bounds; 2. The connection between the approximation radius and the Kolmogorov width is a consequence of a novel duality relationship on the Kolmogorov width, developed by utilizing some deep results from Convex geometry.
Dinesh Manocha - One of the best experts on this subject based on the ideXlab platform.
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incremental penetration depth estimation between Convex Polytopes using dual space expansion
IEEE Transactions on Visualization and Computer Graphics, 2004Co-Authors: Young J Kim, Ming C Lin, Dinesh ManochaAbstract:We present a fast algorithm to estimate the penetration depth between Convex Polytopes in 3D. The algorithm incrementally seeks a "locally optimal solution" by walking on the surface of the Minkowski sums. The surface of the Minkowski sums is computed implicitly by constructing a local dual mapping on the Gauss map. We also present three heuristic techniques that are used to estimate the initial features used by the walking algorithm. We have implemented the algorithm and compared its performance with earlier approaches. In our experiments, the algorithm is able to estimate the penetration depth in about a milli-second on an 1 GHz Pentium PC. Moreover, its performance is almost independent of model complexity in environments with high coherence between successive instances.
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deep dual space expansion for estimating penetration depth between Convex Polytopes
International Conference on Robotics and Automation, 2002Co-Authors: Young J Kim, Ming C Lin, Dinesh ManochaAbstract:We present an incremental algorithm to estimate the penetration depth between Convex Polytopes in 3D. The algorithm incrementally seeks a "locally optimal solution" by walking on the surface of the Minkowski sums. The surface of the Minkowski sums is computed implicitly by constructing a local Gauss map. In practice, the algorithm works well when there is high motion coherence in the environment and is able to compute the optimal solution in most cases.
Frédéric Bosio - One of the best experts on this subject based on the ideXlab platform.
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Real quadrics in Cn, complex manifolds and Convex Polytopes
Acta Mathematica, 2006Co-Authors: Frédéric Bosio, Laurent MeerssemanAbstract:In this paper, we investigate the topology of a class of non-Kahler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics C n which are invariant with respect to the natural action of the real torus (S 1) n onto C n . The quotient space is a simple Convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on Convex Polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple Convex polytope. Finally, as a nice consequence, we obtain that affine non-Kahler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kahler situation.
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Real quadrics in $\Bbb C^n$, complex manifolds and Convex Polytopes
Acta Mathematica, 2006Co-Authors: Frédéric Bosio, Laurent MeerssemanAbstract:In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics in $\Bbb C^n$ which are invariant with respect to the natural action of the real torus $(\Bbb S^1)^n$ onto $\Bbb C^n$. The quotient space is a simple Convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on Convex Polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple Convex polytope. Finally, as a nice consequence, we obtain that affine non Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.