Approximation of convex sets by polytopes

Bronstein, Eli

doi:10.1007/s10958-008-9144-x

Cited by 96 publications

(82 citation statements)

References 139 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As mentioned earlier, it is well known that Ω 1/ε (d−1)/2 facets are required to ε-approximate a Euclidean ball of unit radius (see, e.g., [17]), and this holds for any polytope that that is sufficiently close to a ball in terms of Hausdorff distance. The following utility lemma generalizes this observation to different diameters.…”

Section: Lower Boundmentioning

confidence: 97%

“…An inner ε-approximation is defined similarly but with P ⊆ K. Dudley [31] showed that there exists an outer ε-approximating polytope for any bounded convex body in R d formed by the intersection of O 1/ε (d−1)/2 halfspaces, and Bronshteyn and Ivanov [16] proved an analogous bound on the number of vertices needed to obtain an inner ε-approximation. Both bounds are known to be asymptotically tight in the worst case (see, e.g., [17]). These results have been applied to a number of problems, for example, the construction of coresets [2].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Approximate polytope membership queries

Arya

Fonseca

Mount

2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

View full text Add to dashboard Cite

In the polytope membership problem, a convex polytope K in R d is given, and the objective is to preprocess K into a data structure so that, given any query point q ∈ R d , it is possible to determine efficiently whether q ∈ K. We consider this problem in an approximate setting. Given an approximation parameter ε, the query can be answered either way if the distance from q to K's boundary is at most ε times K's diameter. We assume that the dimension d is fixed, and K is presented as the intersection of n halfspaces. Previous solutions to approximate polytope membership were based on straightforward applications of classic polytope approximation techniques by Dudley (1974) and Bentley et al. (1982). The former is optimal in the worst-case with respect to space, and the latter is optimal with respect to query time.We present four main results. First, we show how to combine the two above techniques to obtain a simple space-time trade-off. Second, we present an algorithm that dramatically improves this trade-off. In particular, for any constant α ≥ 4, this data structure achieves query time roughly O 1/ε (d−1)/α and space roughly O 1/ε (d−1)(1−Ω(log α)/α) . We do not know whether this space bound is tight, but our third result shows that there is a convex body such that our algorithm achieves a space of at least Ω 1/εOur fourth result shows that it is possible to reduce approximate Euclidean nearest neighbor searching to approximate polytope membership queries. Combined with the above results, this provides significant improvements to the best known space-time trade-offs for approximate nearest neighbor searching in R d . For example, we show that it is possible to achieve a query time of roughly O(log n + 1/ε d/4 ) with space roughly O(n/ε d/4 ), thus reducing by half the exponent in the space bound.

show abstract

Section: Lower Boundmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Approximate polytope membership queries

Arya

Fonseca

Mount

2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

View full text Add to dashboard Cite

show abstract

“…Dudley showed that, for ε ≤ 1, any convex body K of unit diameter can be ε-approximated by a convex polytope P with O(1/ε (d−1)/2 ) facets [20]. This bound is known to be tight in the worst case and is achieved when K is a Euclidean ball [16]. Alternatively, Bronshteyn and Ivanov showed the same bound holds for the number of vertices, which is also the best possible [15].…”

Section: Introductionmentioning

confidence: 99%

Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

Arya¹,

Arya²,

Fonseca³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

Convex bodies play a fundamental role in geometric computation, and approximating such bodies is often a key ingredient in the design of efficient algorithms. We consider the question of how to succinctly approximate a multidimensional convex body by a polytope. We are given a convex body K of unit diameter in Euclidean d-dimensional space (where d is a constant) along with an error parameter ε > 0. The objective is to determine a polytope of low combinatorial complexity whose Hausdorff distance from K is at most ε. By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. In the mid-1970's, a result by Dudley showed that O(1/ε (d−1)/2 ) facets suffice, and Bronshteyn and Ivanov presented a similar bound on the number of vertices. While both results match known worst-case lower bounds, obtaining a similar upper bound on the total combinatorial complexity has been open for over 40 years. Recently, we made a first step forward towards this objective, obtaining a suboptimal bound. In this paper, we settle this problem with an asymptotically optimal bound of O(1/ε (d−1)/2 ).Our result is based on a new relationship between ε-width caps of a convex body and its polar. Using this relationship, we are able to obtain a volume-sensitive bound on the number of approximating caps that are "essentially different." We achieve our result by combining this with a variant of the witness-collector method and a novel variable-width layered construction.

show abstract

“…Given

N > 2 n

, but proportional to

n

, we would like to estimate the quantity

\begin{matrix} true \\ right C_{n, N} = sup_{MJX-tex-caligraphicscriptP = {P_{i} : i \in I}} inf ({} \frac{{vol}_{n} (P_{i_{1}} \cap \dots \cap P_{i_{N}})}{{vol}_{n} (\cap MJX-tex-caligraphicscriptP)} : i_{1}, \dots, i_{N} \in I), \end{matrix}

where the supremum is taken over all families

P = {P_{i} : i \in I}

of closed half‐spaces for which

\cap P

has finite and positive volume. A closely related, but not equivalent, question is to give an upper bound for the volume of the smallest polytope

P

with

N

facets that contains a given convex set of volume

1

(for the rich literature in this area, the reader may consult the surveys of Gruber [23] and Bronstein [12]).…”

Section: Introductionmentioning

confidence: 99%

“…where the supremum is taken over all families P = {P i : i ∈ I } of closed halfspaces for which ∩P has finite and positive volume. A closely related, but not equivalent, question is to give an upper bound for the volume of the smallest polytope P with N facets that contains a given convex set of volume 1 (for the rich literature in this area, the reader may consult the surveys of Gruber [23] and Bronstein [12]). Our first result concerns the case of families of symmetric strips in R n .…”

mentioning

confidence: 99%

Brascamp–lieb Inequality and Quantitative Versions of Helly's Theorem

Brazitikos

2016

Mathematika

View full text Add to dashboard Cite

We provide new quantitative versions of Helly's theorem. For example, we show that for every family {Pi:i∈I} of closed half‐spaces in Rn such that P=⋂i∈IPi has positive volume, there exist s⩽αn and i1,…,is∈I such that right center left3ptthickmathspacetrueitalicvolMJX-TeXAtom-ORDn⁡(PMJX-TeXAtom-ORDiMJX-TeXAtom-ORD1∩⋯∩PMJX-TeXAtom-ORDiMJX-TeXAtom-ORDs)⩽(Cn) MJX-TeXAtom-ORDnitalicvolMJX-TeXAtom-ORDn⁡(P), where α,C>0 are absolute constants. These results complement and improve previous work of Bárány et al and Naszódi. Our method combines the work of Srivastava on approximate John's decompositions with few vectors, a new estimate on the corresponding constant in the Brascamp–Lieb inequality and an appropriate variant of Ball's proof of the reverse isoperimetric inequality.

show abstract

Approximation of convex sets by polytopes

Cited by 96 publications

References 139 publications

Approximate polytope membership queries

Approximate polytope membership queries

Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

Brascamp–lieb Inequality and Quantitative Versions of Helly's Theorem

Contact Info

Product

Resources

About