Ugrás a tartalomhoz

Convex Geometry

Csaba Vincze (2013)

University of Debrecen

10. fejezet - Generalized conics

10. fejezet - Generalized conics

The idea of generalization of classical conics is a periodic phenomenon in the history of mathematics. There are lots of points of view of investigations. Generalized conics (especially polyellipses as the additive versions of lemniscates) were investigated from the viewpoint of approximation theory [24], see also [23]. On the other hand these geometric objects appear in optimization problems in a natural way [30], see also [41] and [31]. Another point of view in the literature [47] is the theory of equidistant sets: the United Nations Convention on the Law of the Sea (Article 15) establishes that, in absence of any previous agreement, the delimitation of the territorial sea between countries occurs exactly on the median line every point of which is equidistant of the nearest points to each country. The classical conics can be always realized as sets of equidistant points from two given subsets (circles) in the plane. Here we have another type of generalization together with applications in Minkowski geometry and geometric tomography.

Instead of finite sums integration of the Euclidean distance over the set of foci (curves, surfaces, integration domains) will be used to generalize the classical conics. According to the philosophy of integration (partitions, integral sums) conics defined by this way are just limits of conics in the discrete sense. Therefore it is a purely topological way of the generalization (cf. Weiszfeld's problem and Erdős-Vincze's theorem). There are general difficulties in explicite computations and adaptation to the standard tools in differential geometry such as parametrization and curvature; see e.g. elliptic integrals in case of circular conics in the space. To develop a kind of theory of circular (generalized) conics [56], see also [57], we need subtle estimations [2] and [49] for ellipticintegrals and the Gaussian hypergeometric function. As a similar trend computer simulations, algorithms and estimations can be used instead of the classical tools.

10.1 A panoramic view

Let G be a subset of the Euclidean coordinate space. A generalized conic is a set of points with the same average distance from the points in G. First of all we consider some examples how to calculate such an average distance of a single point from a point-set. The method can be realized in several ways from classical (discrete) means to integration over the set of foci. In most of important cases the common feature of functions measuring the average distance is the convexity. They also satisfy a kind of growth condition. Therefore the level sets are compact convex subsets in the space bounded by compact convex hypersurfaces. They are called generalized conics.In what follows we present some examples and basic facts of the theory of generalized conics.

Example If G is a finite set of points in the space then the average distance can be calculated as the arithmetic mean

F ( p ) : = d ( p , p 1 ) + + d ( p , p m ) m

(10.1)

of distances from the points p(1), ..., p(m) of G. Hypersurfaces of the form F(p)=const. are called polyellipses or polyellipsoids depending on the dimension of the embedding coordinate space. The points p(1), ..., p(m) are called focuses. We have the usual notion of ellipses in case of two different focuses in the coordinate plane.

It is natural to take any other types of mean or their weighted versionsinstead of the standard arithmetic one in formula 10.1. The most important discrete cases are polyellipses with the classical arithmetic mean to calculate the average Euclidean distance from the elements of a finite point-set and lemniscates (with the classical geometric mean to calculate the average Euclidean distance from the elements of a finite point-set). Lemniscates in the plane play a central role in the theory of approximation in the sense that polynomial approximations of holomorphic functions can be interpreted as approximations of curves with lemniscates. In terms of algebra we speak about the roots of polynomials (in terms of geometry we speak about their foci). Endre Vázsonyi posed the problem whether the polyellipses (as the additive versions of lemniscates) have the same approximating property by increasing of the number of the foci or not. The answer is negative (see Erdős-Vincze's theorem [24], see also [60] and chapter 11). To include hyperbolas as a special case of generalized conics we can admit a simple weighted sum of distances instead of means. Parabolas can be given as a special case if not only single points but hyperplanes are also admitted as the element of the set of foci. The pure case of such a construction is presented in the following example.

Example If G isa finite set

H 1 , , H m

(10.2)

of hyperplanes in the coordinate space of dimension n then the average distance can be calculated as the arithmetic mean

F ( p ) : = d ( p , H 1 ) + + d ( p , H m ) m

(10.3)

of distances from the hyperplanes 10.2. Especially let

e 1 : = ( 1,0 , 0 ) , e 2 : = ( 0,1 , 0 , , 0 ) , , e n : = ( 0 , , 0,1 )

be the canonical basis and consider the hyperplanes

H i : = a f f { e 1 , , e i - 1 , e i + 1 , , e n } ,   w h e r e   i = 1 n .

In terms of the coordinates of p we have

F ( p ) = | p 1 | + + | p n | n ,

and hypersurfaces of the form F(p)=const. are just spheres with respect to the 1-norm. They can be also considered as a generalization of conics.

It is a natural question whether how we can calculate the average distance of a single point of the space from a set consisting of infinitely many points. Consider first of all a curve Γ in the space. In order to calculate the average distance of thepoint p from Γ divide the curve into m parts with the same arc-length. After choosing a point from each part let us define the function

F m ( p ) : = d ( p , γ 1 ) + + d ( p , γ m ) m = d ( p , γ 1 ) s 1 + + d ( p , γ m ) s m t h e   a r c - l e n g t h   o f   Γ ,

(10.4)

where s(1)= ... =s(m) is the common arc-length of the pieces and γ(1), ..., γ(m) are points on the curve from the corresponding pieces of the partition. Under reasonable conditions

l i m m F m ( p ) = 1 t h e   a r c - l e n g t h   o f   Γ Γ γ d ( p , γ ) d γ .

In view of this argumentation we can formulate the following definition.

Definition Let Γ be a bounded orientable submanifold[11] in the coordinate space of dimension n with finite positive measure (arc-length, area or volume). The average distance is measured as the integral

F ( p ) : = 1 v o l Γ Γ γ d ( p , γ ) d γ .

(10.5)

Hypersurfaces of the form F(p)=const. are called generalized conics with Γ as the set of focuses.

In this sense generalized conics are "limits" of sequences of polyellipses or polyellipsoids 10.1; cf. Weissfeld's problem of the topological closure of the set of polyellipses in the plane. To generalize the pure case of hyperplanes 10.3 in a similar way we can use the submanifolds of Grassmannians. By taking submanifolds of the product of the coordinate space with Grassmannians or flag manifolds [38] mixed cases can be also presented. Let Γ be a subset of dimension n. The integral

Γ γ d ( p , γ ) d γ

is just the volume of the body C(p) bounded by Γ in the horizontal hyperplane of dimension n and the upper half of the right circular cone with opening angle π/2. It has a vertical axis to the horizontal hyperplane at the vertex p.

Figure 63: The body C(p).

Theorem 10.1.1 The function F is convex satisfying the growth condition

l i m i n f p F ( p ) p > 0 .

(10.6)

Proof Convexity is clear because the integrand is a convex function of the variable p for any fixed element γ in Γ. Since Γ is bounded it is contained in a ball around the origin with a finite radius K. Then

d ( p , γ ) + K d ( p , γ ) + d ( γ , 0 ) d ( p , 0 ) = p

implies the inequality

d ( p , γ ) p 1 - K p .

Integrating both side over Γ

l i m i n f p F ( p ) p = 1 > 0

as was to be stated.

Excercise 10.1.2 Prove that the levels of the function F are bounded.

Hint. Suppose, in contrary, that the level set

L : = { p E n | F ( p ) c }

contains a sequence of points p(1), ..., p(m), ... such that the norms of the elements p(i)'s tend to the infinity. Then

l i m m F ( p m ) p m l i m m c p m = 0

which contradicts to the growth condition 10.6.

Corollary 10.1.3 F has a global minimizer.

Proof The statement follows from the Weierstrass's theorem [13] : if all the level sets of a continuous function defined on a non-empty closed set in the coordinate space of dimension n are bounded then the function has a global minimizer.

Excercise 10.1.4 Prove Weierstrass's theorem: if all the level sets of a continuous functiondefined on a non-empty closed set in the coordinate space of dimension n are bounded then the function has a global minimizer.

Excercise 10.1.5 Under what conditions can we provide the unicity of the minimizer?

Hint. Find a condition for the function F to be strictly convex.