By Howard Eves

From the book's preface:

Since writing the preface of the 1st version of this paintings, the gloomy plight there defined of starting collegiate geometry has brightened significantly. The pendulum turns out certainly to be swinging again and a goodly quantity of good textual fabric is showing.

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**Extra info for A Survey of Geometry (Revised Edition)**

**Sample text**

Discretization of p-stable variables LEMMA: Let 1 < P < 2, e > o. Let 9 be a symmetric p-stable random variable on [0,1], endowed with the Lebesgue measure A, and assume Ilgl\1 = 1. , g" is decreasing and P(g" > t) = P(lgl > t) for all 0 < t < 00) and = g"C£), i = 1, .. :,n. Then there exists an a = a(e,p) such that for all m,n E IN with an, if Yl, ... ,Ym is a sequence of independent, symmetric random variables with each of IYil, i = 1, ... J ' i=l Y= then (1 - e) for all scalars bl , ... , ~ Ibjl ( lip P m ::; ) LbjYj j=l bm.

Since this holds for all A, B as in (*) we get that Ildi+llloo ~ ~ for i = 0, ... , n - l. Applying the lemma we get that if I is a function with Lipschitz constant 1 then (**) If A ~ II n then d(·, A) is a Lipschitz function with constant one so we can use (**). )(log 4)/n. )(log 4)/n) . > -2 On the other hand if P(A) ~ ~ then P(d(·,A) = 0) ~ !. )(log 4)/n. )(log 4)/n. To prove (*) let p be the permutation which changes T with s and leaves the rest at = po 1r then ep(7r)(i) = 7r(i) for all i except possibly for i = j + 1 (where 1r(i) = T,ep(1r)(i) = s) and for i = 7r- 1 (s) (where 7r(i) = sand ep(7r)(i) = po 7r(i) = p(s) = r), place.

Ibjl ( lip P m ::; ) LbjYj j=l bm. (XA is the indicator function of the set A). PROOF: Draw a picture to check that Ilg" - ylll = {I {lin 10 (g" - y)dA ::; 1 g"dA. 0 Let C = C (p) be such that P(g" > t) Then = P(lgl > t) ::; C . t- P • 44 and JIgO - Let yld>' ~ l l/n o ll/n gOd>' ~ Clip. 0 t-l/Pd>. = Cl/p_P_n(l-p)/p . p-1 gl, ... • ,m. Put 1, ... ,m . ,Ym and bn , m ~ clip. ~ 1 . n(l-p)/p. L P j=l ~ = 1, J' = 1, ... Ibj[ j=l Clip. _p_. (m)(p_l)/p. IP ] ]=1 Choose a such that Clip . l!. a(p-l)/p p-l < g.