# Differential Geometry, Part 2 by Chern S., Osserman R. (eds.) By Chern S., Osserman R. (eds.)

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Xn , x1 + x2 + · · · + xn x12 + x22 + · · · + xn2 ≥ , n n with equality if and only if x1 = x2 = · · · = xn . Cauchy–Schwarz Inequality For any real numbers x1 , x2 , . . , xn and y1 , y2 , . . , yn , (x12 + x22 + · · · + xn2 )(y12 + y22 + · · · + yn2 ) ≥ (x1 y1 + x2 y2 + · · · + xn yn )2 , with equality if and only if xi and yi are proportional, i = 1, 2, . . , n. Minkowski’s Inequality For any real numbers x1 , x2 , . . , xn , y1 , y2 , . . , yn , . . , z 1 , z 2 , . . , z n , x12 + y12 + · · · + z 12 + ≥ x22 + y22 + · · · + z 22 + · · · + xn2 + yn2 + · · · + z n2 (x1 +x2 + · · · + xn )2 + (y1 + y2 + · · · + yn )2 + · · · + (z 1 +z 2 + · · · + z n )2 , with equality if and only if xi , yi , .

If λ = 0, then (2) is equivalent to a x− λ 2 b + y− λ 2 = a 2 + b2 − λc . 5. The Tangency Principle 53 This equation deﬁnes: (i) a circle with center O = ( aλ , λb ) if a 2 + b2 − λc > 0; (ii) the point O = ( aλ , λb ) if a 2 + b2 − λc = 0; (iii) the empty set if a 2 + b2 − λc < 0. If λ = 0, then clearly (2) deﬁnes a line if a 2 + b2 > 0, the whole plane if a = b = c = 0, and the empty set if a = b = 0 and c = 0. ♠ One can prove in the same way a space analogue of the above theorem. Note that in the case (a) the level surface L μ is a sphere, a point, or the empty set, whereas in the case (b) it is a plane, the whole space, or the empty set.

Consequently t = π2 + kπ and 2t = π2 + nπ for some integers k and n, which √ √ implies 2 = 2n+1 , a contradiction since 2 is irrational. That is why M N > 1 2k+1 for any t. Chapter 1. Methods for Finding Geometric Extrema 36 We will now show that M N can be made arbitrarily close to 1. For any integer k set tk = π2 + kπ . Then | sin tk | = 1, so at any time tk the point M is at A. To show√that N can be arbitrarily close to A1 at times tk , it is enough to show that | sin( 2tk )| can be arbitrarily close to 1 for appropriate choices of k.