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e(ho0n3
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[SOLVED] Convexity of Open Disc
I'm trying to prove that the open disc in the complex plane given by D = {z : |z - w| < r} is convex.
Let p and q be two points in D. The line segment from p to q is L = {(1 - t)p + tq : 0 <= t <= 1}. Let u be a point an arbitrary point on this segment. If I can show that |u - w| < r, I'm done.
This is essentially a geometry problem. There's probably a proposition in Euclid's Elements that the line segment from any two points on the circle is contained in the circle. How do you show this analytically though?
I'm trying to prove that the open disc in the complex plane given by D = {z : |z - w| < r} is convex.
Let p and q be two points in D. The line segment from p to q is L = {(1 - t)p + tq : 0 <= t <= 1}. Let u be a point an arbitrary point on this segment. If I can show that |u - w| < r, I'm done.
This is essentially a geometry problem. There's probably a proposition in Euclid's Elements that the line segment from any two points on the circle is contained in the circle. How do you show this analytically though?