Question

In Exercises 11 and 12, mark each statement True or False. Justify each answer.

a. If \(y = {c_{\bf{1}}}{{\bf{v}}_{\bf{1}}} + {c_{\bf{2}}}{{\bf{v}}_{\bf{2}}} + {c_{\bf{3}}}{{\bf{v}}_{\bf{3}}}\) and \({c_{\bf{1}}} + {c_{\bf{2}}} + {c_{\bf{3}}} = {\bf{1}}\), then y is a convex combination of \({{\bf{v}}_{\bf{1}}}\), \({{\bf{v}}_{\bf{2}}}\), and \({{\bf{v}}_{\bf{3}}}\).

b. If S is a nonempty set, then conv S contains some points that are not in S.

c. If S and T are convex sets, then \(S \cup T\) is also convex.

Short answer

a. The given statement is False.

b. The given statement is False.

c. The given statement is False.

Step-by-step solution

Check for statement (a)

A combination is said to be convex if all the scalars are positive. It is a necessary condition, which is not mentioned in the given statement.

So, the given statement is false.

Check for statement (b)

By Theorem 10, every point in a nonempty set is \({\mathbb{R}^n}\), then every point in conv S can be represented as a convex combination of fewer points in S. So, S contains some points that are not in S.

So, the given statement is False.

Check for statement (c)

For a convex sets,and a line \(\overline {pq} \) is contained in S. For this \(S \cup T\) is also convex, then \(p,q \in S\). But if, \(p \in S\) and \(q \in T\), then \(\overline {pq} \) is not necessarily in \(S \cup T\).

So, the statement is False.