Ovals and support functions of plane curves

Let $\alpha (s)$ be a closed plane curve. Define the diameter $d_\alpha$ of $\alpha (s)$ to be

Now assume that the curvature $k(s) \geq 1$ for all $s$.

We assume that in the motion defined in problem 103 the point $w=f(z)$ describes in the positive direction a closed curve without double points. This curve is everywhere convex (seen from outside) if and only if for $\left|z\right|=r$

The set of points with respect to which a closed curve is star-shaped forms a convex set. Prove this purely geometric proposition for analytic curves with the help of problem 109.

Let $\mathfrak {R}$ denote a finite convex domain in the $w=u+iv$-plane. For a fixed angle $\varphi$ the expression

assumes a certain maximum $h(\varphi )$ in $\mathfrak {R}$. The function $h(\varphi )$ is periodic with period $2\pi$ and is called the support function of $\mathfrak {R}$. The straight line

is a line of support; its normal points away from $\mathfrak {R}$ and forms the angle $\varphi$ with the positive $u$-axis. If $\mathfrak {R}$ extends to infinity these definitions are modified insofar as a finite maximum exists only in a sector with an opening $\leq \pi$. The two cases: the infinite strip and the half-plane are exceptions; then there exist only two lines of support, or only one, respectively.

Find the support function of the convex domain that consists of the point $a=\left|a\right|e^{i\alpha }$.

The function $w=f(z)$ establishes a one to one correspondence between the disk $\left|z\right|\leq r$ and the convex domain $\mathfrak {R}$. Suppose that $f(z)$ is regular and $f'(z)\neq 0$ at a certain point $z$ on the boundary, $\left|z\right|=r$. In this case a definite line of support (tangent) passes through the boundary point $w=f(z)$ of $\mathfrak {R}$. Express the corresponding quantities $\varphi$ and $h(\varphi )$ in terms of $f(z)$.

The function $w=\log (1+z)$ maps the disk $\left|z\right|=1$ onto an infinite domain contained in the strip $-\frac{\pi }{2}<\mathrm{Im}\left(w\right)<\frac{\pi }{2}$. Its support function (defined only for $-\frac{\pi }{2}\leq \varphi \leq \frac{\pi }{2}$) is

The function $w=\frac{2}{i}\arcsin iz$ maps the disk $\left|z\right|\leq 1$ onto a finite convex domain which has two corners and lies in the strip $-\pi \leq \mathrm{Im}\left(w\right)\leq \pi$. Its support function is

The equation $we^{-w+1}=z$ defines $w=f(z)$. It maps the disk $\left|z\right|\leq 1$ onto a finite convex domain which has one corner and lies in the half-plane $\mathrm{Re}\left(w\right)\leq 1$. Its support function is

in the sector $\frac{\pi }{4}<\varphi <\frac{7\pi }{4}$ it is given in parametric form

whereby $w$ describes the boundary, $\left|we^{-w+1}\right|=1$.

Let $h(\varphi )$ denote the support function of the infinite convex domain considered in problem 114. The sequence $f_{1}(z),f_{2}(z),\ldots ,f_{n}(z),\ldots$

where $z=re^{i\varphi }$, is bounded in the entire half-plane $\mathrm{Re}\left(z\right)\geq 0$ [problems 12, and Part II, problem 220]; $h(\varphi )$ is the smallest function of the angle $\varphi$ that keeps the sequence bounded.

Let $h(\varphi )$ denote the support function of the convex domain considered in problem 115. The sequence $f_{1}(z),f_{2}(z),\ldots ,f_{n}(z),\ldots$

where $z=re^{i\varphi }$, is bounded in the entire $z$-plane [problems 13, and Part II, problem 221]; $h(\varphi )$ is the smallest function of the angle $\varphi$ that keeps the sequence bounded.

Let $h(\varphi )$ denote the support function of the convex domain considered in problem 116. Then

$z=re^{i\varphi }$, in the entire plane.

On a closed convex curve with continuous curvature there are always at least three different pairs of points with the following property: The tangents at the two points of a pair are parallel and the curvature of the curve at the two points is the same.

Let the boundary $\partial D$ of a domain $D$ consist of one smooth closed curve with parametric equation $z = z(t), a \leqslant t \leqslant b$, where the function $z(t)$ has, on the segment $[a, b]$, a continuous and nonzero derivative. Prove that: