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Algebraic Geometry - Bowdoin 1985, Part 1 by Bloch S. (ed.)

By Bloch S. (ed.)

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25) From the definition of the signs of the curvatures k(t) and kd (t) it is not difficult to deduce that for 1 − kd > 0 the signs of k(t) and kd (t) coincide, but for 1 − kd < 0 these signs are opposite. 25) the second statement of the theorem follows. 1. 2 takes the form Rd = R − d. 23. Let γ (t) (−∞ < a ≤ t ≤ b < ∞) be a regular parameterization of a curve γ of the class C 2 . Prove that if |d| < inf t∈[a,b] 1 , k(t) then γd ∪ γ−d can be defined as a set of points whose distances from γ are equal to |d|.

Inscribe in a curve γ a closed polygonal line σ with the vertices A1 , A2 , . . , An , An+1 (An+1 = A1 ) such that the integral curvature of every arc γi = Ai Ai+1 of γ is not greater than π . On each arc γi take a point Bi where the tangent line is parallel to the straight line Ai Ai+1 . Denote by αi the inner angle of the polygonal line σ at the vertex Ai . Then γ¯i k(s) ds = π −αi , where γ¯i is the arc of γ from Bi to Bi+1 . Consequently, n γ On the other hand, k(s) ds = i=1 n i=1 γ n γ¯i k(s) ds = nπ − αi .

Solution. 1. 6. If a simple closed curve has a nonnegative curvature at each of its points, then it is convex. 14. 6. Solution. Assume that γ is not convex. Then there exist two points A and B on γ such that the line segment AB lies outside of D(γ ), and γ is located on one side of the straight line AB. The points A and B divide γ onto two arcs, γ1 and γ2 . One of the curves σ1 = γ1 ∪ AB and σ2 = γ2 ∪ AB contains D(γ ). Assume that this curve is σ2 . Find a point P on the arc γ1 with the maximal distance from the straight line AB.

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