# Tagged Questions

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### Prove that if $A, B$ are convex , $B$ is closed, $C$ is bounded and $A+C \subset B+C$ then $A\subset B$

Given the sets $A,B,C \in \mathbb{R}^n$ such that: $$A+C \subset B+C$$ Show that if $A,B$ are convex, $B$ is closed and $C$ is bounded then $A\subset B$. I kind of understand the geometrical ...
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### Is closure of convex subset of $X$ is again a convex subset of $X$?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...
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### How to show convexity of a ball in metric space?

If $(X,\|\cdot\|)$ is a normed linear space, then how to show any ball $B(x,r)$ is convex? I know that if $x,y\in A\subset V$ then $[x,y]\subset A$, where $A$ is a convex subset of vector space $V$ ...
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### How to show that $\mathbb R^n$ with the $1$-norm is not isometric to $\mathbb R^n$ with the infinity norm for $n>2$?

Could you please give me a hint to prove that $\mathbb{R}^n$ with the 1-norm $\lvert x\rvert_1=\lvert x_1\rvert+\cdots+\lvert x_n\rvert$ is not isometric to $\mathbb{R}^n$ with the infinity-norm ...
I wonder how to prove the following statement. Let $V$ be a $d$-dimensional normed space with $d \geq 3$, let $P \subset V$ be a $(d-2)$-dimensional polytope. Then there is an $\epsilon > 0$ such ...
How to prove that the set of extreme points of $B_{\ell^1} = \{v \in \ell^1 : \| v \| \le 1\}$ is $\{ +e^N, -e^N : N=1,2,3,\ldots \}$, where $e^N$ denotes the Nth standard basis element in $\ell_1$: ...