Page 249 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 5. A convex plane curve :[a,b]   is strictly convex if k  0. Prove that if :[a,b]  is
2
2

,
a strictly convex simple closed curve, then for every   there is a unique t [a,b] such
1

that e (t) = .
1
6. Let :[0,L]   be a strictly convex simple closed curve. The width w(t) of  at t [0,L]
2
is the distance between the tangent line at (t) and the tangent line at the unique point


 (t ) satisfying e (t )   e (t) . A curve has constant width if w is independent of t. Prove

1
1
that if  has constant width then:
(a) The line between (t) and (t ) is perpendicular to the tangent lines at those points.



(b) The curve  has length L = w.
7. Let :[0,L]   be a simple closed curve. By the Jordan Curve Theorem, the complement
2
of  has two connected components, one of which is bounded. The area enclosed by  is the
area of this component, and according to Green’s Theorem, it is given by:
A    x dy    xy dt,

where the orientation is chosen so that the normal e points into the bounded component.
2
Let L be the length of , and let  be a circle of width 2r equal to some width of . Prove:
(a) A  1   xy  yx dt.
2

(b) A + r  Lr.
2
(c) The isoperimetric inequality: 4A  L .
2
(d) If equality holds in (3) then  is a circle.
8. Prove that if a convex simple closed curve has four vertices, then it cannot meet any circle
in more than four points.

Answers: Self Assessment

1. parametrized curve
2. unit tangent
3. Frenet frame equations

4. motion of  3

20.9 Further Readings

Books Ahelfors, D.V. : Complex Analysis
Conway, J.B. : Function of one complex variable
Pati, T. : Functions of complex variable
Shanti Narain : Theory of function of a complex Variable

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