Question

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

a. A set is convex if \({\bf{x}},{\bf{y}} \in S\) implies that the line segment between x and y is contained is S.

b. If S and T are convex sets, then \(S \cap T\) is also convex.

c. If S is a nonempty subset of \({\mathbb{R}^{\bf{5}}}\) and \({\bf{y}} \in {\bf{conv}}\,\,S\), then there exists distinct points \({{\bf{v}}_{\bf{1}}}\),….\({{\bf{v}}_{\bf{6}}}\).

Short answer

a. The given statement is True.

b. The given statement is True

c. The given statement is True.

Step-by-step solution

Check for statement (a)

Using the definition of convex, a set S is convex if for each x, yin S, the line \(\overline {xy} \) is contained in S.

So, the given statement is True.

Check for statement (b)

From theorem 8, if S and Trepresent a convex set, their intersection is also a convex set.

So, the given statement is True.

Check for statement (c)

According to theorem 10, every point in the set in \({\mathbb{R}^n}\), then every point in conv S can be represented as a convex combination. So, y can be expressed as,

\(y = {c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ..... + {c_6}{{\bf{v}}_6}\)y=

It is a convex combination of \({{\bf{v}}_1}\), \({{\bf{v}}_2}\),….,\({{\bf{v}}_6}\).

So, the statement is True.