Download Companion to Concrete Mathematics: Vol. I: Mathematical by By (author) Z. A. Melzak PDF

By By (author) Z. A. Melzak

A two-volume therapy in one binding, this supplementary textual content stresses intuitive charm and ingenuity. It employs actual analogies, encourages challenge formula, and offers problem-solving tools. 1973 and 1976 variants.

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First, we may assume that C is convex, for otherwise a perturbation may be applied which keeps L constant and increases A. This perturbation is illustrated in Figure 2 and consists of reflecting a concavity outward by mirroring it in a tangent line. Next, let P and Q be any two points on our convex curve Cwhich divide the circumference into two parts of equal length. Then it may be assumed that the area of C is also divided by the line PQ into two equal parts. For otherwise one half of the curve together with its reflection in PQ will enclose a larger area FI GURE 2.

To which an answer may be given: yes, X is real but the equations describing it are almost always approximations, and thus these equations describe not X but some hypothetical simplified Xl; therefore the need for existence-proofs follows from modesty of admitting one's partial ignorance. ---------------------------~-s p Q FIGURE 3. Area perturbation. 8. ISOPERIMETRIC PROBLEMS FOR CONVEX HULLS 23 Returning to the isoperimetric problem of the circle, we observe that what has been proved is this: if there exists a maximizing curve then it must be a circle.

Then we have L r(-::'+/)) 2 FIGURE 4. Isodiametry. and therefore A(X)=! {12 r2(O)dO=! {12[r2(o)+r2(_::+o)] dO:o:;::. 2 -1[/2 2 0 2 4 Hence A(X) attains its maximum n/4 when X is the circular disk of unit diameter. We can formulate a general isodiametric problem as follows. Let X be a set in the Euclidean n-space and n(X) its diameter. X) = AaF(X), F(X)::;; F( Y) if X c;: Y, F(X) is continuous. Here AX is the scaled-up replica of X in the ratio }, : I (and A > 0). a is a positive constant, and in (d) the continuity is taken with respect to the Hausdorff set-distance d(X, Y): if X, is the union of all balls of radius a centered in X, then d(X, Y) = infra: X c Y, and Y c XJ.

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