Approximation Of Convex Bodies Research Articles

Floating bodies and approximation of convex bodies by polytopes

We describe some results on approximation of convex bodies by polytopes. Best and random approximations are considered and compared. The geometric quantities related to the convex body, that appear naturally in such approximation questions are the affine surface areas. Those, and their relation to floating bodies, will be discussed as well. In this survey we give only a very selective collection of results on approximation by polytopes.

Approximations of convex bodies by measure-generated sets

Given a Borel measure $$\mu $$ on $$\mathbb {R}^{n}$$ , we define a convex set by $$\begin{aligned} \mathrm{M}\left( \mu \right) =\bigcup _{\begin{array}{c} 0\le f\le 1, \int _{\mathbb {R}^{n}}f\,\mathrm{d}\mu =1 \end{array} }\left\{ \int _{\mathbb {R}^{n}}yf\left( y\right) \,\mathrm{d}\mu \left( y\right) \right\} , \end{aligned}$$ where the union is taken over $$\mu $$ -measurable functions $$f:\mathbb {R}^{n}\rightarrow \left[ 0,1\right] $$ such that $$\int _{\mathbb {R}^{n}}f\,\mathrm{d}\mu =1$$ and $$\int _{\mathbb {R}^{n}}yf\left( y\right) \,\mathrm{d}\mu \left( y\right) $$ exists. We study the properties of these measure-generated sets, and use them to investigate natural variations of problems of approximation of general convex bodies by polytopes with as few vertices as possible. In particular, we study an extension of the vertex index which was introduced by Bezdek and Litvak. As an application, we prove that for any non-degenerate probability measure $$\mu $$ , one has the lower bound $$\begin{aligned} \int _{\mathbb {R}^{n}}\left\| x\right\| _{Z_{1}\left( \mu \right) }\,\mathrm{d}\mu \left( x\right) \ge c\sqrt{n}, \end{aligned}$$ where $$Z_{1}\left( \mu \right) $$ is the $$L_{1}$$ -centroid body of $$\mu $$ .

On the Approximation of Convex Bodies by Ellipses with Respect to the Symmetric Difference Metric

Given a centrally symmetric convex body \(K \subset \mathbb {R}^d\) and a positive number \(\lambda \), we consider, among all ellipsoids \(E \subset \mathbb {R}^d\) of volume \(\lambda \), those that best approximate K with respect to the symmetric difference metric, or equivalently that maximize the volume of \(E\cap K\): these are the maximal intersection (MI) ellipsoids introduced by Artstein-Avidan and Katzin. The question of uniqueness of MI ellipsoids (under the obviously necessary assumption that \(\lambda \) is between the volumes of the John and the Loewner ellipsoids of K) is open in general. We provide a positive answer to this question in dimension \(d=2\). Therefore we obtain a continuous 1-parameter family of ellipses interpolating between the John and the Loewner ellipses of K. In order to prove uniqueness, we show that the area \(I_K(E)\) of the intersection \(K \cap E\) is a strictly quasiconcave function of the ellipse E, with respect to the natural affine structure on the set of ellipses of area \(\lambda \). The proof relies on smoothening K, putting it in general position, and obtaining uniform estimates for certain derivatives of the function \(I_K(\mathord {\cdot })\). Finally, we provide a characterization of maximal intersection positions, that is, the situation where the MI ellipse of K is the unit disk, under the assumption that the two boundaries are transverse.

Approximation of convex bodies by multiple objective optimization and an application in reachable sets

In this paper, we focus on approximating convex compact bodies. For a convex body described as the feasible set in objective space of a multiple objective programme, we show that finding it is equivalent to finding the non-dominated set of a multiple objective programme. This equivalence implies that convex bodies can be approximated using multiple objective optimization algorithms. Therefore, we propose a revised outer approximation algorithm for convex multiple objective programming problems to approximate convex bodies. Finally, we apply the algorithm to solve reachable sets of control systems and use numerical examples to show the effectiveness of the algorithm.

Approximation of convex bodies by polytopes with respect to minimal width and diameter

Denote by ${\mathcal K}^d$ the family of convex bodies in $E^d$ and by $w(C)$ the minimal width of $C \in {\mathcal K}^d$. We ask for the greatest number $\Lambda_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ contains a polytope $P$ with at most $n$ vertices for which $\Lambda_n ({\mathcal K}^d) \leq \frac{w(P)}{w(C)}$. We give a lower estimate of $\Lambda_n ({\mathcal K}^d)$ for $n \geq 2d$ based on estimates of the smallest radius of $\big\lfloor {\frac{n}{2}} \big\rfloor$ antipodal pairs of spherical caps that cover the unit sphere of $E^d$. We show that $\Lambda_3 ({\mathcal K}^2) \geq {\frac 1 2}(3- \sqrt 3)$, and $\Lambda_n ({\mathcal K}^2) \geq \cos {\frac \pi {2 \lfloor {n/2} \rfloor}}$ for every $n \geq 4$. We also consider the dual question of estimating the smallest number $\Delta_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ there exists a polytope $P \supset C$ with at most $n$ facets for which $\frac{{\rm diam}(P)}{{\rm diam}(C)} \leq \Delta_n ({\mathcal K}^d)$. We give an upper bound of $\Delta_n ({\mathcal K}^d)$ for $n \geq 2d$. In particular, $\Delta_n ({\mathcal K}^2) \leq 1/ \cos {\frac \pi {2 \lfloor {n/2} \rfloor}}$ for $n \geq 4$.

Asymptotic properties of the estimate refinement method in polyhedral approximation of multidimensional balls

The estimate refinement method for the polyhedral approximation of convex compact bodies is considered. In the approximation of convex bodies with a smooth boundary, this method is known to generate polytopes with an optimal order of growth of the number of vertices and facets depending on the approximation error. The properties of the method are examined as applied to the polyhedral approximation of a multidimensional ball. As vertices of approximating polytopes, the method is shown to generate a deep holes sequence on the surface of the ball. As a result, previously obtained combinatorial properties of convex hulls of the indicated sequences, namely, the convergence rates with respect to the number of faces of all dimensions and the optimal growth of the cardinality of the facial structure (of the norm of the f-vector) can be extended to such polytopes. The combinatorial properties of the approximating polytopes generated by the estimate refinement method are compared to the properties of polytopes with a facial structure of extremal cardinality. It is shown that the polytopes generated by the method are similar to stacked polytopes, on which the minimum number of faces of all dimensions is attained for a given number of vertices.

Optimal growth order of the number of vertices and facets in the class of Hausdorff methods for polyhedral approximation of convex bodies

The internal polyhedral approximation of convex compact bodies with twice continuously differentiable boundaries and positive principal curvatures is considered. The growth of the number of facets in the class of Hausdorff adaptive methods of internal polyhedral approximation that are asymptotically optimal in the growth order of the number of vertices in approximating polytopes is studied. It is shown that the growth order of the number of facets is optimal together with the order growth of the number of vertices. Explicit expressions for the constants in the corresponding bounds are obtained.

Approximation of convex bodies by inscribed simplices of maximum volume

The Banach-Mazur distance between an arbitrary convex body and a simplex in Euclidean n-space En is at most n + 2. We obtain this estimate as an immediate consequence of our theorem which says that for an arbitrary convex body C in En and for any simplex S of maximum volume contained in C the homothetical copy of S with ratio n + 2 and center in the barycenter of S contains C. In general, this ratio cannot be improved, as it follows from the example of any double-cone.

Rate of Approximation of Regular Convex Bodies by Convex Algebraic Level Surfaces

We consider the problem of the approximation of regular convex bodies in ℝ d by level surfaces of convex algebraic polynomials. Hammer (in Mathematika 10, 67–71, 1963) verified that any convex body in ℝ d can be approximated by a level surface of a convex algebraic polynomial. In Jaen J. Approx. 1, 97–109, 2009 and subsequently in J. Approx. Theory 162, 628–637, 2010 a quantitative version of Hammer’s approximation theorem was given by showing that the order of approximation of convex bodies by convex algebraic level surfaces of degree n is \(\frac{1}{n}\). Moreover, it was also shown that whenever the convex body is not regular (that is, there exists a point on its boundary at which the convex body possesses two distinct supporting hyperplanes), then \(\frac{1}{n}\) is essentially the sharp rate of approximation. This leads to the natural question whether this rate of approximation can be improved further when the convex body is regular. In this paper we shall give an affirmative answer to this question. It turns out that for regular convex bodies a o(1/n) rate of convergence holds. In addition, if the body satisfies the condition of C 2-smoothness the rate of approximation is \(O(\frac{1}{n^{2}})\).

Approximating the convex Edgeworth-Pareto hull in integer multi-objective problems with monotone criteria

A method for the iterative polyhedral approximation of the convex Edgeworth-Pareto hull is proposed and examined experimentally. This method is designed for integer multi-objective problems with monotone objective functions and constraints given by a computational module. It is based on a synthesis of the ideas of the branch-and-bound method and the methods for the polyhedral approximation of convex bodies. A sequence of interior and exterior polyhedral sets is constructed so as to approximate the Edgeworth-Pareto hull to the desired accuracy. The results of the theoretical and experimental analyses of the proposed method are presented.

The initial convergence rate of adaptive methods for polyhedral approximation of convex bodies

The convergence rate at the initial stage is analyzed for a previously proposed class of asymptotically optimal adaptive methods for polyhedral approximation of convex bodies. Based on the results, the initial convergence rate of these methods can be evaluated for arbitrary bodies (including the case of polyhedral approximation of polytopes) and the resources sufficient for achieving optimal asymptotic properties can be estimated.

Duality theory of optimal adaptive methods for polyhedral approximation of convex bodies

A duality theory is developed to describe iterative methods for polyhedral approximation of convex bodies. The various types of approximation problems requiring the application of the duality theory are considered. Based on the theory, approximation methods can be designed for bodies with a dual description (in terms of the support/distance function) and methods can be developed that are optimal in terms of dual complexity characteristics of approximating polytopes (vertices/facets). New optimal methods based on the theory are formulated.

Erratum to “Approximation of convex bodies by axially symmetric bodies”

Erratum to “Approximation of convex bodies by axially symmetric bodies”

Approximation of convex bodies by axially symmetric bodies

Let C C be an arbitrary planar convex body. We prove that C C contains an axially symmetric convex body of area at least 2 3 | C | \frac {2}{3}|C| . Also approximation by some specific axially symmetric bodies is considered. In particular, we can inscribe a rhombus of area at least 1 2 | C | \frac {1}{2}|C| in C C , and we can circumscribe a homothetic rhombus of area at most 2 | C | 2|C| about C C . The homothety ratio is at most 2 2 . Those factors 1 2 \frac {1}{2} and 2 2 , as well as the ratio 2 2 , cannot be improved.

Approximation of Convex Bodies by Centrally Symmetric Bodies

We present an analog of the well-known theorem of F. John about the ellipsoid of maximal volume contained in a convex body. Let C be a convex body and let D be a centrally symmetric convex body in the Euclidean d-space. We prove that if D′ is an affine image of D of maximal possible volume contained in C, then C a subset of the homothetic copy of D′ with the ratio 2d-1 and the homothety center in the center of D′. The ratio 2d-1 cannot be lessened as a simple example shows.

The cross-section body, plane sections of convex bodies and approximation of convex bodies, I

For a convex body K⊂ℝ d we investigate three associated bodies, its intersection body IK (for 0∈int K), cross-section body CK, and projection body IIK, which satisfy IK⊂CK⊂IIK. Conversely we prove CK⊂const1(d)I(K−x) for some x∈int K, and IIK⊂const2 (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L⊂ℝ d a convex body, we take n random segments in L, and consider their ‘Minkowski average’ D. We prove that, for V(L) fixed, the supremum of V(D) (with also n∈N arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M⊂ℝ d a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1≤p<∞) also are minimized, for V(L) fixed, by the ellipsoids. For k=2, the supremum (n∈N arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.

Centres of Convex Sets inLpMetrics

It is shown that for each convex bodyA⊂Rnthere exists a naturally defined family GA⊂C(Sn−1) such that for everyg∈GA, and every convex functionf:R→Rthe mappingy↦∫Sn−1f(g(x)−⦠y, x⦔)dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toLpmetrics is discussed.

Approximation of convex bodies by simplices

This paper deals with the following kind of approximation of a convex bodyQ in Euclidean space E n by simplices: which is the smallest positive numberh S(Q) such thatS 1 ⊂Q ⊂S 2 for a simplexS 1 and its homothetic copyS 2 of ratioh S(Q). It is shown that ifS 0 is a simplex of maximal volume contained inQ, then a homothetic copy ofS 0 of ratio 13/3 containsQ.

Approximation of convex bodies by rectangles

For every plane convex body there is a pair of inscribed and circumscribed homothetic rectangles. The positive ratio of homothety is not greater than 2.

Approximation of Convex Bodies by Triangles

We show that for every plane convex body $C$ there exist a triangle ${T_1}$ and its image ${T_2}$ under a homothety with ratio $\tfrac {5} {2}$ such that ${T_1} \subset C \subset {T_2}$. We prove the conjecture of Grünbaum that if $C$ is centrally symmetric, then ${T_1},{T_2}$ can be chosen so that their centroids coincide with the center of $C$.