By Luke Y.L.

ISBN-10: 0124599400

ISBN-13: 9780124599406

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C P. As a subset of I!!. 5. 2. Polar polytopes and quotient polytopes 37 (a) The subset F* of P* is a face. 5. One inequality is obvious because of F* C n(aff F) and the dimension formula: dim F* ~ dim n(aff F) = n - 1 - dim F. There remains the inequality"::,:". 11, there exist facets Fj such that F = n~=o F j • Since 0 E int P, we can find vectors u j =I 0 with Fj = P n Hu j . We may assume that t = codim F; then, the u j are affinely independent. If we can show that u j E Fj (c F*), for every j, then, dim F* ::': t - 1 = n - 1 - dim F follows.

The support function h K is linear if and only if K is a point. 3. Show explicitly that d K, dK" in Examples 3 and 4, are norms. 4. Characterize those convex bodies K for which ddx + y) = ddx) implies that x and y are multiples. + dK(y) 6. 1I with respect to the unit sphere S := {x (x, x) = I}. 1I with fj. 1I , then, ° ° rr(u) = Hu := {x I (x, u) = I}. If the affine subspaces U and V which generate W are not parallel and if W does not contain 0, then, rr(W) = rr(U) n rr(V). Note that rr 0 rr is the identity.

We choose u =f. 0 such that K C H-. Then, (xo, u) = sUPxeK(x, u) = hKCu) which implies (b). 4 Definition. A function and x, y E ]R", f(h f : ]R" + (1 ~ ]R is said to be convex if, for all 0 - ),,)y) ~ Vex) K FIGURE 9a,b. + (1 - ),,)f(y)· :s )" :s 1 20 I Convex Bodies Note that if f is convex and L is an affine subspace ofJR", then, flL is also convex. 1 and x, y E JR, the graph r (f) of a convex function f lies "below" the line-segment [(x, f (x», (y, f (y»] in JR2 • Hence for convex f, if a :::: -1 < b < 0, feb) = 1, and f(O) = 0, then, (a, f(a) and (-b, - f(-b» are "above" the line through (b, l) and (0,0), so that f(a) ~ and feb) ~ Example 2.

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Algorithms for computations of mathematical functions by Luke Y.L.


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