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Codes, metrics and topologies using barcode creation for none control to generate, create none image in none pdf-417 generating T h e o r e m 1 none none .12.1 Let (X, d) be a complete metric space and let H(X) denote the space of nonempty compact subsets of X.

Let x X and let B H(X). Then there exists at least one point b = b(x) B such that d(x, b) d(x, b(x)) for all b B. Proof Fix x X.

Then the function f : B X R de ned by f (b) = d(x, b) for all b B is continuous and B is compact. Hence there exists at least one point in B where the value of f is a minimum. We denote such a point by b B.

Notice that b may change when x changes, so we write b = b(x). Theorem 1.12.

1 enables us to make the following de nition. D e f i n i t i o n 1.12.

2 Let (X, d) be a complete metric space. Let H(X) denote the space of nonempty compact subsets of X. Then the distance from a point x X to B H(X) is de ned by D B (x) := min{d(x, b) : b B}.

We refer to D B (x) as the shortest-distance function of the set B. E x e r c i s e 1.12.

3 Let X = = {(x, y) R2 : 0 x 1, 0 y 1}. Let dmax ((x1 , y1 ), (x2 , y2 )) = max{. Microsoft Office Official Website x1 x2 , . y1 y2 }. Let B = {(x, none for none y) : x 2 + y 2 = 0.25}.

Calculate D B ((0.6, 0.8)).

E x e r c i s e 1.12.4 Show that D B (x) d(x, y) + D B (y) for all x, y X.

Use this to show that D B (x) is a continuous function of x X. E x e r c i s e 1.12.

5 Prove that if C, D H(X) with C D then DC (x) D D (x) for all x H(X). For given d 0 we call the set of points L d := {x X : D B (x) = d} a level set of D B (x). All points on L d are at the same distance d from B.

In R2 these level sets {L d : d 0} may form a graceful family of curves, like patterns of ripples, shaped like B, produced by simultaneous disturbances on a water surface or like the wavefronts of light at successive equally spaced time intervals after tiny coherent light pulses are emitted by the points of B at an initial time. We can imagine optical devices, based on the latter idea, that generate approximate level sets of D B (x) when B R2 . For example, schematically, we can imagine a collection of light-emitting diodes organized in two dimensions to form a discrete model for B.

We suppose that these diodes are turned on and off rapidly,. 1.12 The Hausdorff metric Figure 1.26 The none none space around a fern image is painted using different colours for different level curves of the shortest-distance function DF (x , y ). These level curves do not possess well-de ned tangents at all their points.

Also, the perturbation P 0 , a small purple disk, makes no difference to the shortest-distance function close to the fern but modi es it further away.. while an array of ultra-fast and sensitive charge-coupled devices, something like the CCD chip in a digital camera, in the same plane as the diodes is used to photograph the wavefront at different times. In Figure 1.26 we show for comparison D F (x) and D F P0 (x), where F is a fernlike subset of R2 and P0 R2 is a small disk.

From left to right: the subset F R2 and a small disk P0 ; some level curves of D F (x); some level curves of D F P0 (x) (the outermost contour, red, contains points equidistant from F and P0 ); the same as the preceding image but more contours are shown. We see that D F (x) = D F P0 (x) whenever D F (x) is suf ciently small but that P0 provides a serious perturbation to the shortest-distance function at points suf ciently far away from F, in some directions. In Figure 1.

27 we show a close-up of the level sets of D F (x) in the vicinity of the subset F R2 . It is fascinating to imagine these lovely patterns at higher resolutions. In Figure 1.

28 we show an arti cial artistic work. It was made using the shortest-distance function associated with the euclidean metric. Four objects were drawn and coloured, then level sets of the shortest-distance function for the coloured points were computed and rendered.

Paths of steepest descent In this subsection we continue to discuss shortest-distance functions. Let B H(R2 ). At those points (x, y) R2 where the shortest-distance function D B (x, y) is differentiable, grad D B (x, y) = D B D B , x y.

is a vector poi none for none nting along the path of steepest descent from x to the nearest point on B. When the underlying metric is the euclidean metric, and the level sets of D B (x, y) are differentiable curves, this vector is oriented perpendicular to the level set through the point (x, y). In this case, paths of steepest descent for.

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