arXiv subjects

Sunil Arya

Publications and source records attributed to Sunil Arya.

13 recordsLinked to original sources

arXiv2026

Duality Bounds for Convexified Packing in Hilbert Geometry

Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$, where $0 \in \text{int} G$ and $G \subset \text{int} K$. Given $\alpha > 0$, the Hilbert packing number $M_H(G, K; \alpha)$ is the maximum cardinality of a set of points in $G$, each pair of which is separated by a distance of at least $\alpha$ in the Hilbert geometry defined by $K$. The Hilbert convexified packing number $\widehat{M}_H(G, K; \alpha)$ is the maximum length of a sequence of points in $G$, such that each point is separated by distance at least $\alpha$ from the convex hull of its predecessors. We prove a dimension-free primal-polar bound for convexified packing in Hilbert geometry. Letting $G^\circ$ and $K^\circ$ denote the polar bodies, we show that there exist absolute constants $C, c > 0$ such that, for every $\alpha > 0$, \[ \widehat{M}_H(G, K; \alpha) ~ \leq ~ C \cdot \widehat{M}_H(K^\circ, G^\circ; c\alpha)^2 M_H(K^\circ, G^\circ; c\alpha). \] As a direct corollary, we have \[ \widehat{M}_H(G, K; \alpha) ~ \leq ~ C \cdot M_H(K^\circ, G^\circ; c\alpha)^3. \] Thus, the primal convexified packing number is bounded by a fixed polynomial in the ordinary packing number for the reversed polar bodies, with absolute constants independent of the dimension. This is motivated by the duality conjecture for packing and covering numbers, which relates covering $G$ by $K$ to covering $K^\circ$ by $G^\circ$. Our results represent a key step in extending the work of Artstein, Milman, Szarek, and Tomczak-Jaegermann from normed spaces to Hilbert geometries.

math.MG↗

arXiv2026

On the Duality of Coverings in Hilbert Geometry

We prove polarity duality for covering problems in Hilbert geometry. Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$ where $G \subset \operatorname{int}(K)$ and $\operatorname{int}(G)$ contains the origin. Let $N^H_K(G,\alpha)$ and $S^H_K(G,\alpha)$ denote, respectively, the minimum numbers of radius-$\alpha$ Hilbert balls in the geometry induced by $K$ needed to cover $G$ and $\partial G$. Our main result is a Hilbert-geometric analogue of the K\"{o}nig-Milman covering duality: there exists an absolute constant $c \geq 1$ such that for any $\alpha \in (0,1]$, \[ c^{-d}\,N^H_{G^{\circ}}(K^{\circ},\alpha) ~ \leq ~ N^H_K(G,\alpha) ~ \leq ~ c^{d}\,N^H_{G^{\circ}}(K^{\circ},\alpha), \] and likewise, \[ c^{-d}\,S^H_{G^{\circ}}(K^{\circ},\alpha) ~ \leq ~ S^H_K(G,\alpha) ~ \leq ~ c^{d}\,S^H_{G^{\circ}}(K^{\circ},\alpha). \] We also recover the classical volumetric duality for translative coverings of centered convex bodies, and obtain a new boundary-covering duality in that setting. The Hilbert setting is subtler than the translative one because the metric is not translation invariant, and the local Finsler unit ball depends on the base point. The proof involves several ideas, including $\alpha$-expansions, a stability lemma that controls the interaction between polarity and expansion, and, in the boundary case, a localized relative isoperimetric argument combined with Holmes--Thompson area estimates. In addition, we provide an alternative proof of Faifman's polarity bounds for Holmes--Thompson volume and area in the Funk and Hilbert geometries.

math.MG↗

arXiv2026

Cauchy's Surface Area Formula in the Funk Geometry

Cauchy's surface area formula expresses the surface area of a convex body as the average area of its orthogonal projections over all directions. While this tool is fundamental in Euclidean geometry, with applications ranging from geometric tomography to approximation theory, extensions to non-Euclidean settings remain less explored. In this paper, we establish an analog of Cauchy's formula for the Funk geometry induced by a convex body $K$ in $\mathbb{R}^d$, for the Holmes--Thompson surface area. The formula is based on central projections to boundary points of $K$. We show that when $K$ is a convex polytope, the formula reduces to a weighted sum of contributions associated with the vertices of $K$. Finally, as a consequence of our analysis, we derive a generalization of Crofton's formula for surface areas in the Funk geometry. By viewing Euclidean, Minkowski, Hilbert, and hyperbolic geometries as limiting or special cases of the Funk setting, our results provide a unified framework for these classical surface area formulas.

cs.CG↗

arXiv2023

Approximate Nearest Neighbor Searching with Non-Euclidean and Weighted Distances

We present a new approach to approximate nearest-neighbor queries in fixed dimension under a variety of non-Euclidean distances. We are given a set $S$ of $n$ points in $\mathbb{R}^d$, an approximation parameter $\varepsilon > 0$, and a distance function that satisfies certain smoothness and growth-rate assumptions. The objective is to preprocess $S$ into a data structure so that for any query point $q$ in $\mathbb{R}^d$, it is possible to efficiently report any point of $S$ whose distance from $q$ is within a factor of $1+\varepsilon$ of the actual closest point. Prior to this work, the most efficient data structures for approximate nearest-neighbor searching in spaces of constant dimensionality applied only to the Euclidean metric. This paper overcomes this limitation through a method called convexification. For admissible distance functions, the proposed data structures answer queries in logarithmic time using $O(n \log (1 / \varepsilon) / \varepsilon^{d/2})$ space, nearly matching the best known bounds for the Euclidean metric. These results apply to both convex scaling distance functions (including the Mahalanobis distance and weighted Minkowski metrics) and Bregman divergences (including the Kullback-Leibler divergence and the Itakura-Saito distance).

cs.CG↗

arXiv2023

Optimal Area-Sensitive Bounds for Polytope Approximation

Approximating convex bodies is a fundamental problem in geometry. Given a convex body $K$ in $\mathbb{R}^d$ for a fixed dimension $d$, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff error $\varepsilon$. The best known uniform bound, due to Dudley (1974), shows that $O((\text{diam}(K)/\varepsilon)^{(d-1)/2})$ facets suffice. Although this bound is optimal for fat objects, such as Euclidean balls, it is far from optimal for ``skinny'' convex bodies. Skinniness can be characterized relative to the Euclidean ball. Given a convex body $K$, define its area radius, $\text{arad}(K)$, to be the radius of the Euclidean ball having the same surface area as $K$. It follows from generalizations of the isoperimetric inequality that $\text{diam}(K) \geq 2 \cdot \text{arad}(K)$. We show that, given a convex body whose minimum width is at least $\varepsilon$, it is possible to approximate the body by a polytope having $O((\text{arad}(K)/\varepsilon)^{(d-1)/2})$ facets. Our approach works by first reducing the problem of approximating convex bodies to that of approximating convex functions. We employ a classical concept from convexity, called Macbeath regions. We demonstrate that there is a polar relationship between the Macbeath regions of a function and the Macbeath regions of its Legendre dual. This is combined with known bounds on the Mahler volume to bound the total size of the approximation.

cs.CG↗

arXiv2023

Optimal Volume-Sensitive Bounds for Polytope Approximation

Approximating convex bodies is a fundamental question in geometry, which has a wide variety of applications. Given a convex body $K$ in $\textbf{R}^d$ for fixed $d$, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff error $\varepsilon$. It is known that $O((\text{diam}(K)/\varepsilon)^{(d-1)/2})$ facets suffice and are necessary for many instances, such as the Euclidean ball. However, this bound is far from optimal for ``skinny'' convex bodies. A natural way to characterize the skinniness of a convex object is in terms of its relationship to the Euclidean ball. Given a convex body $K$, its \emph{volume diameter} $\Delta_d(K)$ is defined to be the diameter of a Euclidean ball of the same volume as $K$. The \emph{surface diameter} $\Delta_{d-1}(K)$ is defined analogously for surface area. It follows from generalizations of the isoperimetric inequality that $\text{diam}(K) \geq \Delta_{d-1}(K) \geq \Delta_d(K)$. Arya, da Fonseca, and Mount proved that the diameter-based bound could be made sensitive to the surface diameter, improving the above bound to $O((\Delta_{d-1}(K)/\varepsilon)^{(d-1)/2})$. In this paper, we strengthen this by proving the existence of an approximation with $O((\Delta_d(K)/\varepsilon)^{(d-1)/2})$ facets. As a function of volume alone, this bound is tight up to constant factors. Our improvements arise from a combination of new ideas. We exploit known properties of the original body and its polar dual. In order to obtain a volume-sensitive bound, we explore the problem of computing a low-complexity polytope that is sandwiched between two given convex bodies. We show that this problem can be reduced to a covering problem involving a natural intermediate body based on the harmonic mean. Our proof relies on a geometric analysis of a relative notion of fatness involving these bodies.

cs.CG↗

arXiv2023

Economical Convex Coverings and Applications

Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body $K$ and $\epsilon> 0$, a covering is a collection of convex bodies whose union covers $K$ such that a constant factor expansion of each body lies within an $\epsilon$ expansion of $K$. Coverings have been employed in many applications, such as approximations for diameter, width, and $\epsilon$-kernels of point sets, approximate nearest neighbor searching, polytope approximations, and approximations to the Closest Vector Problem (CVP). It is known how to construct coverings of size $n^{O(n)} / \epsilon^{(n-1)/2}$ for general convex bodies in $\textbf{R}^n$. In special cases, such as when the convex body is the $\ell_p$ unit ball, this bound has been improved to $2^{O(n)} / \epsilon^{(n-1)/2}$. This raises the question of whether such a bound generally holds. In this paper we answer the question in the affirmative. We demonstrate the power and versatility of our coverings by applying them to the problem of approximating a convex body by a polytope, under the Banach-Mazur metric. Given a well-centered convex body $K$ and an approximation parameter $\epsilon> 0$, we show that there exists a polytope $P$ consisting of $2^{O(n)} / \epsilon^{(n-1)/2}$ vertices (facets) such that $K \subset P \subset K(1+\epsilon)$. This bound is optimal in the worst case up to factors of $2^{O(n)}$. As an additional consequence, we obtain the fastest $(1+\epsilon)$-approximate CVP algorithm that works in any norm, with a running time of $2^{O(n)} / \epsilon ^{(n-1)/2}$ up to polynomial factors in the input size, and we obtain the fastest $(1+\epsilon)$-approximation algorithm for integer programming. We also present a framework for constructing coverings of optimal size for any convex body (up to factors of $2^{O(n)}$).

cs.CG↗

arXiv2019

Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

This paper considers the question of how to succinctly approximate a multidimensional convex body by a polytope. Given a convex body $K$ of unit diameter in Euclidean $d$-dimensional space (where $d$ is a constant) and an error parameter $\varepsilon > 0$, the objective is to determine a convex polytope of low combinatorial complexity whose Hausdorff distance from $K$ is at most $\varepsilon$. By combinatorial complexity we mean the total number of faces of all dimensions. Classical constructions by Dudley and Bronshteyn/Ivanov show that $O(1/\varepsilon^{(d-1)/2})$ facets or vertices are possible, respectively, but neither achieves both bounds simultaneously. In this paper, we show that it is possible to construct a polytope with $O(1/\varepsilon^{(d-1)/2})$ combinatorial complexity, which is optimal in the worst case. Our result is based on a new relationship between $\varepsilon$-width caps of a convex body and its polar body. 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 main result by combining this with a variant of the witness-collector method and a novel variable-thickness layered construction of the economical cap covering.

cs.CG↗

arXiv2018

Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums

Approximation problems involving a single convex body in $d$-dimensional space have received a great deal of attention in the computational geometry community. In contrast, works involving multiple convex bodies are generally limited to dimensions $d \leq 3$ and/or do not consider approximation. In this paper, we consider approximations to two natural problems involving multiple convex bodies: detecting whether two polytopes intersect and computing their Minkowski sum. Given an approximation parameter $\varepsilon > 0$, we show how to independently preprocess two polytopes $A,B$ into data structures of size $O(1/\varepsilon^{(d-1)/2})$ such that we can answer in polylogarithmic time whether $A$ and $B$ intersect approximately. More generally, we can answer this for the images of $A$ and $B$ under affine transformations. Next, we show how to $\varepsilon$-approximate the Minkowski sum of two given polytopes defined as the intersection of $n$ halfspaces in $O(n \log(1/\varepsilon) + 1/\varepsilon^{(d-1)/2 + α})$ time, for any constant $α> 0$. Finally, we present a surprising impact of these results to a well studied problem that considers a single convex body. We show how to $\varepsilon$-approximate the width of a set of $n$ points in $O(n \log(1/\varepsilon) + 1/\varepsilon^{(d-1)/2 + α})$ time, for any constant $α> 0$, a major improvement over the previous bound of roughly $O(n + 1/\varepsilon^{d-1})$ time.

cs.CG↗

arXiv2017

Approximate Polytope Membership Queries

In the polytope membership problem, a convex polytope $K$ in $\mathbb{R}^d$ is given, and the objective is to preprocess $K$ into a data structure so that, given any query point $q \in \mathbb{R}^d$, it is possible to determine efficiently whether $q \in K$. We consider this problem in an approximate setting. Given an approximation parameter $\varepsilon$, the query can be answered either way if the distance from $q$ to $K$'s boundary is at most $\varepsilon$ 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 $α\ge 4$, this data structure achieves query time $O(1/\varepsilon^{(d-1)/α})$ and space roughly $O(1/\varepsilon^{(d-1)(1 - O(\log α)/α)})$. 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 $Ω( 1/\varepsilon^{(d-1)(1-O(\sqrtα)/α} )$. Our 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 $\mathbb{R}^d$.

cs.CG↗

arXiv2017

Near-Optimal $\varepsilon$-Kernel Construction and Related Problems

The computation of (i) $\varepsilon$-kernels, (ii) approximate diameter, and (iii) approximate bichromatic closest pair are fundamental problems in geometric approximation. In this paper, we describe new algorithms that offer significant improvements to their running times. In each case the input is a set of $n$ points in $R^d$ for a constant dimension $d \geq 3$ and an approximation parameter $\varepsilon > 0$. We reduce the respective running times (i) from $O((n + 1/\varepsilon^{d-2})\log(1/\varepsilon))$ to $O(n \log(1/\varepsilon) + 1/\varepsilon^{(d-1)/2+α})$, (ii) from $O((n + 1/\varepsilon^{d-2})\log(1/\varepsilon))$ to $O(n \log(1/\varepsilon) + 1/\varepsilon^{(d-1)/2+α})$, and (iii) from $O(n / \varepsilon^{d/3})$ to $O(n / \varepsilon^{d/4+α}),$ for an arbitrarily small constant $α> 0$. Result (i) is nearly optimal since the size of the output $\varepsilon$-kernel is $Θ(1/\varepsilon^{(d-1)/2})$ in the worst case. These results are all based on an efficient decomposition of a convex body using a hierarchy of Macbeath regions, and contrast to previous solutions that decompose space using quadtrees and grids. By further application of these techniques, we also show that it is possible to obtain near-optimal preprocessing time for the most efficient data structures to approximately answer queries for (iv) nearest-neighbor searching, (v) directional width, and (vi) polytope membership.

cs.CG↗

arXiv2016

On the Combinatorial Complexity of Approximating Polytopes

Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body $K$ of diameter $\mathrm{diam}(K)$ is given in Euclidean $d$-dimensional space, where $d$ is a constant. Given an error parameter $\varepsilon > 0$, the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from $K$ is at most $\varepsilon \cdot \mathrm{diam}(K)$. By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that $O(1/\varepsilon^{(d-1)/2})$ facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is $\tilde{O}(1/\varepsilon^{(d-1)/2})$, where $\tilde{O}$ conceals a polylogarithmic factor in $1/\varepsilon$. This is a significant improvement upon the best known bound, which is roughly $O(1/\varepsilon^{d-2})$. Our result is based on a novel combination of both old and new ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of Bárány and Larman's economical cap covering. Finally, we use a deterministic adaptation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.

cs.CG↗

arXiv2016

Optimal Approximate Polytope Membership

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 a query point $q \in R^d$, it is possible to determine efficiently whether $q \in K$. We consider this problem in an approximate setting and assume that $d$ is a constant. Given an approximation parameter $\varepsilon > 0$, the query can be answered either way if the distance from $q$ to $K$'s boundary is at most $\varepsilon$ times $K$'s diameter. Previous solutions to the problem were on the form of a space-time trade-off, where logarithmic query time demands $O(1/\varepsilon^{d-1})$ storage, whereas storage $O(1/\varepsilon^{(d-1)/2})$ admits roughly $O(1/\varepsilon^{(d-1)/8})$ query time. In this paper, we present a data structure that achieves logarithmic query time with storage of only $O(1/\varepsilon^{(d-1)/2})$, which matches the worst-case lower bound on the complexity of any $\varepsilon$-approximating polytope. Our data structure is based on a new technique, a hierarchy of ellipsoids defined as approximations to Macbeath regions. As an application, we obtain major improvements to approximate Euclidean nearest neighbor searching. Notably, the storage needed to answer $\varepsilon$-approximate nearest neighbor queries for a set of $n$ points in $O(\log \frac{n}{\varepsilon})$ time is reduced to $O(n/\varepsilon^{d/2})$. This halves the exponent in the $\varepsilon$-dependency of the existing space bound of roughly $O(n/\varepsilon^d)$, which has stood for 15 years (Har-Peled, 2001).

cs.CG↗