Exercise 10.7

(a) First we need to show that $Q^k$ is convex. Let $\check{x},\check{y}\in Q^k$, so that the components satisfy

Let $0\le \lambda \le 1$, and let $ž=\lambda \check{x}+(1-\lambda )\check{y}$. Then

so that $z_i$ lies between $x_i$ and $y_i$, and $\sum <!--\; FUNCTION\; APPLICATION\; -->z_i$ lies between $\sum <!--\; FUNCTION\; APPLICATION\; -->x_i$ and $\sum <!--\; FUNCTION\; APPLICATION\; -->y_i$. Hence $ž\in Q^k$.

Let $C$ be a convex subset of $k$ containing $\check{0},{ě}_1,\dots ,{ě}_k$; we need to show that $Q^k\subset C$. We can consider $Q^i\subset Q^j$ for $i<j$ by letting the components with index greater than $i$ be 0. I am going to show that $Q^i\subset C$, $i=1,\dots ,k$ by induction. Let $\check{x}\in Q^1$. Then $\check{x}=x_1{ě}_1+(1-x_1)\check{0}$ for $0\le x_1\le 1$, so that $\check{x}\in C$. Now suppose that $Q^{i-1}\subset C$ and let $\check{x}\in Q^i$. Then $x_1+\dots +x_i\le 1$ implies

so that

Hence $\check{x}=(1-x_i){\check{x}}^{\prime }+x_i{ě}_i\in C$ since $0\le x_i\le 1$, which shows that $Q^i\subset C$.

(b) Let $X,Y$ be vector spaces and let $\check{f}=\check{f}(\check{0})+A$, for some $A\in L(X,Y)$, be an affine mapping from $X$ to $Y$. Let $C$ be a convex subset of $X$, and let ${\check{y}}_1=\check{f}({\check{x}}_1)$, ${\check{y}}_2=\check{f}({\check{x}}_2)$ be elements of $\check{f}(C)$ for some ${\check{x}}_1\in C$ and ${\check{x}}_2\in C$. Then for $0\le \lambda \le 1$, we have $\lambda {\check{x}}_1+(1-\lambda ){\check{x}}_2\in C$, so that

Hence $\check{f}(C)$ is convex.