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40
On Volume and Edge Lengths of Simplices in Normed Spaces
, 2008
"... Let Md be a ddimensional normed space with norm ‖ · ‖ and let B be the unit ball in Md. Let us fix a Lebesgue measure VB in M d with VB(B) = 1. This measure will play the role of the volume in Md. We consider an arbitrary simplex T in Md with prescribed edge lengths. For the case d = 2, sharp up ..."
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Let Md be a ddimensional normed space with norm ‖ · ‖ and let B be the unit ball in Md. Let us fix a Lebesgue measure VB in M d with VB(B) = 1. This measure will play the role of the volume in Md. We consider an arbitrary simplex T in Md with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of VB(T) are determined. For the case d ≥ 3 it is noticed that the tight lower bound of VB(T) is zero.
A generalization of Steinhagen’s theorem
 ABH. MATH. SEM. UNIV. HAMBURG
, 1993
"... The theorem of Steinhagen establishes a relation between inradius and width of a convex set. The half of the width can be interpreted as the minimum of inradii of all 1dimensional orthogonal projections of a convex set. By considering idimensional projections we obtain series of idimensional inr ..."
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Cited by 5 (3 self)
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The theorem of Steinhagen establishes a relation between inradius and width of a convex set. The half of the width can be interpreted as the minimum of inradii of all 1dimensional orthogonal projections of a convex set. By considering idimensional projections we obtain series of i
A geometric analysis of subspace clustering with outliers
 ANNALS OF STATISTICS
, 2012
"... This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information a ..."
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Cited by 61 (3 self)
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This paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes. As is common in computer vision or unsupervised learning applications, we do not know in advance how many subspaces there are nor do we have any information about their dimensions. We develop a novel geometric analysis of an algorithm named sparse subspace clustering (SSC) [11], which significantly broadens the range of problems where it is provably effective. For instance, we show that SSC can recover multiple subspaces, each of dimension comparable to the ambient dimension. We also prove that SSC can correctly cluster data points even when the subspaces of interest intersect. Further, we develop an extension of SSC that succeeds when the data set is corrupted with possibly overwhelmingly many outliers. Underlying our analysis are clear geometric insights, which may bear on other sparse recovery problems. A numerical study complements our theoretical analysis and demonstrates the effectiveness of these methods.
The primes contain arbitrarily long polynomial progressions
 Acta Math
"... Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε ..."
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Cited by 47 (7 self)
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Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε such that x+P1(m),..., x+Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case Pi = (i − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties. Contents
Coincidences of simplex centers and related facial structures
, 2004
"... Abstract. This is an investigation of the geometric properties of simplices in Euclidean ddimensional space for which analogues of the classical triangle centers coincide. A presentation of related results is given, partially unifying known results for d = 2 and d = 3. ..."
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Cited by 6 (4 self)
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Abstract. This is an investigation of the geometric properties of simplices in Euclidean ddimensional space for which analogues of the classical triangle centers coincide. A presentation of related results is given, partially unifying known results for d = 2 and d = 3.
Tube formulas and complex dimensions of selfsimilar tilings
 Acta Appl. Math., arXiv:math/0605527v5 [math.DS] inria00538956, version 1  24 Nov 2010
"... Abstract. We use the selfsimilar tilings constructed in [32] to define a generating function for the geometry of a selfsimilar set in Euclidean space. This tubular zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mapp ..."
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Cited by 25 (13 self)
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Abstract. We use the selfsimilar tilings constructed in [32] to define a generating function for the geometry of a selfsimilar set in Euclidean space. This tubular zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the selfsimilar tiling as the poles of the tubular zeta function and hence develop a tube formula for selfsimilar tilings inR d. The resulting power series inεis a fractal extension of Steiner’s classical tube formula for convex bodies K⊆R d. Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i=0, 1,...,d−1, just as Steiner’s does. However, our formula also contains a term for each complex dimension. This provides further justification for the term “complex dimension”. It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on
STABLE COMPLEXITY AND SIMPLICIAL VOLUME OF MANIFOLDS
"... Abstract. Let the ∆complexity σ(M) of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the appropriate infimum on all finite coverings of M we can promote σ to a multiplicativ ..."
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Cited by 8 (1 self)
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Abstract. Let the ∆complexity σ(M) of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the appropriate infimum on all finite coverings of M we can promote σ to a
MULTIDIMENSIONAL SOFIC SHIFTS WITHOUT SEPARATION AND THEIR FACTORS
"... Abstract. For d ≥ 2 we exhibit mixing Z d shifts of finite type and sofic shifts with large entropy but poorly separated subsystems (in the sofic examples, the only minimal subsystem is a single point). These examples consequently have very constrained factors; in particular, no nontrivial full shi ..."
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Cited by 15 (6 self)
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Abstract. For d ≥ 2 we exhibit mixing Z d shifts of finite type and sofic shifts with large entropy but poorly separated subsystems (in the sofic examples, the only minimal subsystem is a single point). These examples consequently have very constrained factors; in particular, no nontrivial full shift is a factor. We also provide examples to distinguish certain mixing conditions, and develop the natural class of “block gluing ” shifts. In particular, we show that block gluing shifts factor onto all full shifts of strictly smaller entropy. 1.
A Bratteli Diagram for Commuting Homeomorphisms of the Cantor Set
 Internat. J. Math
, 1999
"... This paper presents a construction of a sequence of KakutaniRohlin Towers and a corresponding simple Bratteli Diagram, B, for a general minimal aperiodic Z action on a Cantor Set, X, with d 2. This is applied to the description of the orbit structure and to the calculation of the ordered gro ..."
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Cited by 11 (1 self)
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This paper presents a construction of a sequence of KakutaniRohlin Towers and a corresponding simple Bratteli Diagram, B, for a general minimal aperiodic Z action on a Cantor Set, X, with d 2. This is applied to the description of the orbit structure and to the calculation of the ordered group of coinvariants. For each minimal continuous Z action on X there is a minimal continuous Z action on X which has the same orbits on the set X nY , where Y is of the form [ v2Z , where Y is closed and nowhere dense, and Y is null with respect to every probability measure invariant under the action. Also the ordered top cohomology of (X; Z ), H , is a simple dimension group. There is a positive order embedding surjection K 0 (B) \Gamma! H which becomes an order isomorphism upon quotienting by the infinitessimals. So there is no ordered cohomological obstruction to continuous orbit equivalence between minimal Z actions and Z actions in general.
Results 1  10
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