Question

In \({\mathbb{R}^{\bf{2}}}\), let \(S = \left\{ {\left( {\begin{aligned}{{}{}}{\bf{0}}\\y\end{aligned}} \right):{\bf{0}} \le y < {\bf{1}}} \right\} \cup \left\{ {\left( {\begin{aligned}{{}{}}{\bf{2}}\\{\bf{0}}\end{aligned}} \right)} \right\}\). Describe (or sketch) the convex hull of S.

Short answer

\({k_1}{v_1} + {k_2}{v_2} + ...... + {k_n}{v_n} + \left( 0 \right){v_{n + 1}}\)

Step-by-step solution

Step 1:Find constants for convex of the hull

Let \({c_1}\), \({c_2}\),…., \({c_n}\)be constants(\({c_i} \ge 0,\,\forall i\)). Alsofor some constants\({k_1}\), \({k_2}\),….,\({k_n}\),

\({k_1} = \frac{{{c_1}}}{{{c_1} + {c_2} + .... + {c_n}}},\,\,{k_2} = \frac{{{c_2}}}{{{c_1} + {c_2} + .... + {c_n}}}\,,..............,{k_n} = \frac{{{c_n}}}{{{c_1} + {c_2} + .... + {c_n}}}\).

And

\({c_1} + {c_2} + ...... + {c_n} = 1\).

For the above constants,

\(\begin{aligned}{}{k_1} + {k_2} + .... + {k_n} &= \frac{{{c_1}}}{{{c_1} + {c_2} + ... + {c_n}}} + ..... + \frac{{{c_n}}}{{{c_1} + {c_2} + ... + {c_n}}}\\ &= \frac{{{c_1} + {c_2} + ..... + {c_n}}}{{{c_1} + {c_2} + ..... + {c_n}}}\\ &= 1\end{aligned}\)

Find the convex hull

The given vector sets are:

\({v_1} = \left( {\begin{aligned}{{}{}}0\\{{y_1}}\end{aligned}} \right)\), \({v_2} = \left( {\begin{aligned}{{}{}}0\\{{y_2}}\end{aligned}} \right)\),…………,\({v_n} = \left( {\begin{aligned}{{}{}}0\\{{y_n}}\end{aligned}} \right)\), \({v_{n + 1}} = \left( {\begin{aligned}{{}{}}2\\0\end{aligned}} \right)\)

Therefore,theconvex hullhas elements in it, which is given by the expression:

\({k_1}{v_1} + {k_2}{v_2} + ...... + {k_n}{v_n} + \left( 0 \right){v_{n + 1}}\)

So, the vector \({\bf{y}}\) is \(2{{\bf{v}}_1} - \frac{3}{2}{{\bf{v}}_2} + \frac{1}{2}{{\bf{v}}_3}\).