Question

Describe the convex hull of the set S of points \(\left( {\begin{aligned}{ {20}{c}}x\\y\end{aligned}} \right)\) in \({\mathbb{R}^{\bf{2}}}\) that satisfy the given conditions. Justify your answers. (Show that an arbitrary point p in S belongs to conv S.)

a. \(y = \frac{1}{x}\) and \(x \ge \frac{{\bf{1}}}{{\bf{2}}}\)

b. \(y = {\bf{sin}}x\)

c. \(y = {x^{\frac{{\bf{1}}}{{\bf{2}}}}}\) and \(x \ge {\bf{0}}\)

Short answer

a. \(\left( {\begin{aligned}{ {20}{c}}{\frac{1}{2} + a - \frac{a}{{2x}}}\\{2 - \frac{{2a}}{x} + \frac{a}{{{x^2}}}}\end{aligned}} \right)\)

b. \(\left( {\begin{aligned}{ {20}{c}}{x + 2n\pi t}\\{\sin x}\end{aligned}} \right)\)

c. \(t\left( {\begin{aligned}{ {20}{c}}x\\{\sqrt x }\end{aligned}} \right)\)

Step-by-step solution

Find the convex hull for condition (a)

The given condition is as follows:

\(y = \frac{1}{x}\)and \(x \ge \frac{1}{2}\).

The convex S all points \({\bf{p}}\) of the form:

\(\begin{aligned}{c}{\bf{p}} = \left( {1 - t} \right)\left( {\begin{aligned}{ {20}{c}}{\frac{1}{2}}\\2\end{aligned}} \right) + t\left( {\begin{aligned}{ {20}{c}}x\\{\frac{1}{x}}\end{aligned}} \right)\\ = \left( {\begin{aligned}{ {20}{c}}{\frac{1}{2} + t\left( {x - \frac{1}{2}} \right)}\\{2 - t\left( {2 - \frac{1}{x}} \right)}\end{aligned}} \right)\end{aligned}\)

Let \(t = \frac{a}{x}\), then it is represented as below:

\(\begin{aligned}{c}{\bf{p}} = \left( {\begin{aligned}{ {20}{c}}{\frac{1}{2} + \frac{a}{x}\left( {x - \frac{1}{2}} \right)}\\{2 - \frac{a}{x}\left( {2 - \frac{1}{x}} \right)}\end{aligned}} \right)\\ = \left( {\begin{aligned}{ {20}{c}}{\frac{1}{2} + a - \frac{a}{{2x}}}\\{2 - \frac{{2a}}{x} + \frac{a}{{{x^2}}}}\end{aligned}} \right)\end{aligned}\)

The figure below represents the sketch.

It can be observed from the figure that \(\mathop {\lim }\limits_{x \to \infty } {\bf{p}}\left( x \right) = \left( {\begin{aligned}{ {20}{c}}{\frac{1}{2} + a}\\2\end{aligned}} \right)\) is close to the line \(y = 2\). It can also be seen that the curve \(y = \frac{1}{x}\) is in \({\rm{conv}}\,\,S\) for \(x \ge \frac{1}{2}\).

Find the convex hull for condition (b)

The given condition is as follows:

\(y = \sin x\)

\(\sin \left( {\sin x} \right)\)is a cyclic function,

\(\sin \left( {x + 2n\pi } \right) = \sin x\)

So, the convex hull is as follows:

\(\begin{aligned}{c}{\bf{p}} = \left( {1 - t} \right)\left( {\begin{aligned}{ {20}{c}}x\\{\sin x}\end{aligned}} \right) + t\left( {\begin{aligned}{ {20}{c}}{x + 2n\pi }\\{\sin \left( {x + 2n\pi } \right)}\end{aligned}} \right)\\ = \left( {\begin{aligned}{ {20}{c}}{x + 2n\pi t}\\{\sin x}\end{aligned}} \right)\end{aligned}\)

The figure below represents the sketch for \({\bf{p}}\).

As \( - 1 \le \sin x \le 1\), for x and a, a real number r can be selected so that \(r = x + 2n\pi t\) (\(0 \le t \le 1\)).

Find the convex hull for condition (c)

The given condition is as follows:

\(y = {x^{\frac{1}{2}}}\)

So, the convex hull is as follows:

\(\begin{aligned}{c}{\bf{p}} = \left( {1 - t} \right)\left( {\begin{aligned}{ {20}{c}}0\\0\end{aligned}} \right) + t\left( {\begin{aligned}{ {20}{c}}x\\{\sqrt x }\end{aligned}} \right)\\ = t\left( {\begin{aligned}{ {20}{c}}x\\{\sqrt x }\end{aligned}} \right)\end{aligned}\)

The figure below represents a sketch of p.

For \(t = \frac{a}{x}\),

\(\mathop {\lim }\limits_{x \to \infty } {\bf{p}} = \left( {\begin{aligned}{ {20}{c}}a\\0\end{aligned}} \right)\).

The above equation shows that points are close to the line \(y = 0\) in \({\rm{conv}}\,\,S\).