Question

A subset $S \subset \mathbb{R}^n$ is called convex if, for all $\mathbf{x}, \mathbf{y} \in S$, the line segment joining $\mathbf{x}$ to $\mathbf{y}$ is also in $S$, i.e., $t \mathbf{x}+(1-t) \mathbf{y} \in S$ for all $0 \leq t \leq 1$. Prove that the unit ball is a convex subset of a normed vector space. Is the unit sphere convex?

    A subset $S \subset \mathbb{R}^n$ is called convex if, for all $\mathbf{x}, \mathbf{y} \in S$, the line segment joining $\mathbf{x}$ to $\mathbf{y}$ is also in $S$, i.e., $t \mathbf{x}+(1-t) \mathbf{y} \in S$ for all $0 \leq t \leq 1$. Prove that the unit ball is a convex subset of a normed vector space. Is the unit sphere convex?

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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 3, Problem 30

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Step 1

- The unit ball $B$ in a normed vector space $V$ is defined as $B = \{\mathbf{x} \in V : \|\mathbf{x}\| \leq 1\}$. - The unit sphere $S$ in a normed vector space $V$ is defined as $S = \{\mathbf{x} \in V : \|\mathbf{x}\| = 1\}$. Show more…

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A subset $S \subset \mathbb{R}^n$ is called convex if, for all $\mathbf{x}, \mathbf{y} \in S$, the line segment joining $\mathbf{x}$ to $\mathbf{y}$ is also in $S$, i.e., $t \mathbf{x}+(1-t) \mathbf{y} \in S$ for all $0 \leq t \leq 1$. Prove that the unit ball is a convex subset of a normed vector space. Is the unit sphere convex?

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