Harmonic Analysis and Convex Geometry, Fall 2013, November 15.

Instructor: Dmitry Ryabogin

Assignment 10.

1. Problem 1.

a) Find the curvature of the curve

x = t − sin t, y = 1 − cos t, z = 4 sin \frac{t}{2}.

b) Find the curvature of the curve given implicitly by the equations

x + sinh x = sin y + y, z + e^{z} = x + log(1 + x) + 1

at the point (0, 0, 0).

2. Problem 2. Find the expression for the curvature of the plane curve given in polar coordinates.

3. Problem 3. Let the convex curve α be a boundary of a convex body K ⊂ ℝ^{2}.

a) Prove that the length of α is given by L = ∫_{0}^{2π}h(θ)dθ, where h(θ) := h_{K}((cos θ, sin θ)).

Hint: Problem 2 b), Assignment 7.

b) Prove that if K is of constant breadth (width) μ, then length(α) = πμ. Conclude that all bodies of the given constant breadth have the same perimeter.

c) Prove that the line x cos θ + y sin θ = h(θ) is tangent to the curve at the point

α(θ) = (x(θ), y(θ)) = (h(θ) cos θ − h′(θ) sin θ, h(θ) sin θ + h′(θ) cos θ).

Hint: A regular convex plane curve can be defined as an envelope of its tangent lines. Differentiate x cos θ + y sin θ = h(θ) with respect to θ and solve the system.

(In general, a regular curve α is an envelope of the family of curves F if at every point α is tangent to at least one of the curves in F and for every segment of α there are infinitely many points in the segment where α is tangent to curves from F. Use (prove, if you want) the fact that for every point (x, y) of the envelope of the family F = {γ_{β}}_{β∈(a,b)} of curves (given by the equations φ(x, y, β) = 0) there exists β ∈ (a, b) such that

φ(x, y, β) = 0, \frac{∂}{∂β} φ(x, y, β) = 0 ).

d) Prove that the radius of curvature of the curve at α(θ) is R(θ) = h(θ) + h′′(θ).

Hint: Differentiate again

dx(θ) = −(h(θ) + h′′(θ)) sin θdθ, dy(θ) = (h(θ) + h′′(θ)) cos θdθ.

and observe that

(x′(θ), y′(θ)) · (− sin θ, cos θ) = \sqrt{(x′(θ))^{2} + (y′(θ))^{2}} = \frac{ds}{dθ}.

e) Conclude that the length of α is “an average” of the radius of curvature L =

∫_{0}^{2π}R(θ)dθ.

f) Prove that the area enclosed by α is given by A = \frac{1}{2} ∫_{0}^{2π}h(θ)R(θ)dθ. Hint: Use the formula A = \frac{1}{2} ∫_{0}^{2π}(x(θ)y′(θ) − x′(θ)y(θ))dθ from Calc III.

g) Prove that the area enclosed by α is given by A = \frac{1}{2} ∫_{0}^{2π}(h(θ)^{2} − h′(θ)^{2})dθ. Hint: Integrate by parts.