2
votes
0answers
47 views

Geometric conditions equivalent to a set being the unit circle for some norm

Here's the question, as in the textbook (Real Mathematical Analysis, Pugh). The unit ball with respect to a norm $||\, \cdot \,||$ on $\mathbb{R}^2$ is $$ \{ v \in \mathbb{R}^2 : ||\, v \,|| ...
1
vote
1answer
79 views

Proving the inequality of Cauchy-Schwarz in an Euclidean space. [duplicate]

It says let (G, <.,.>) be an euclidean space. Show that for all x, y belonging to G: modulus<x,y> <= sqrt<x,x> * sqrt<y,y> and in the ...
0
votes
1answer
103 views

Which is the correct definition of stationary point for real-valued functions in Euclidean space?

Given a multivariable real-valued function $f$ whose first partials all exist (but which aren't all continuous) at $p$, it is possible that $f$ is not (totally) differentiable at $p$. But since the ...
0
votes
1answer
131 views

Which is the correct Hessian matrix (the standard matrix of a bilinear form)?

Please note the typo in the first entry: $\frac {\partial^2f} {\partial x_1\partial x_2}$ should instead be $\frac{\partial^2f} {\partial x_1\partial x_1}$. Also, this Hessian matrix need not be ...
6
votes
2answers
129 views

Circle Chord Sequence

This is my first post, so be nice! When I was in my first Geometry class in high school, I asked the teacher the following: Given a circle of radius 2a, find the length of the chord running parallel ...
6
votes
3answers
195 views

Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space?

I have question. Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space ? Please can you give me an advice some book names? Thank you!
2
votes
1answer
147 views

prove that the sphere with a hair in $IR^{3}$ is not locally Euclidean at q. Hence it cannot be a topological manifold.

A fundamental theorem of topology, the theorem on invariance of dimension, states that if two nonempty open sets $U ⊂ R_{n}$ and $V ⊂ R_{m}$ are homeomorphic, then n = m. prove that the sphere with a ...
1
vote
1answer
62 views

Please give an example in $ \mathbb{R}^{n} $ of a set that satisfies the upper bound in the Kuratowski Closure-Complement Theorem.

Please give an example in $ \mathbb{R}^{n} $ of a set $ T $ that satisfies the upper bound of $ 14 $ in the Kuratowski Closure-Complement Theorem. Thanks!
6
votes
1answer
89 views

Orthogonality between three vectors whose coordinates are nondecreasing

Say that a vector $x=(x_1,x_2, \ldots ,x_n)\in {\mathbb R}^n$ is nondecreasing if $x_1 \leq x_2\leq \ldots \leq x_n$. Can anyone show or find a counterexample to the following : if three nondecreasing ...
1
vote
0answers
58 views

Proof by vectors segments, Euclidean.

Hi I have a midterm this Friday, and our prof gave us this practice midterm, I got $6$ of the $10$ questions but couldn't do the last $4$, could you please help me out in this question, I have no idea ...
-3
votes
2answers
164 views

Just next to a certain real number, there is a real number. [closed]

{ReekMaths will clarify the question later.Please do nothing with the present question.}A straight line in 2 dimensional geometry is composed of infinitely many points. According to this, we assume ...
3
votes
1answer
38 views

Is there a name for this “euclidean” set of real numbers?

Let $E$ be the smallest set of real numbers such that: (1) $1\in E$, (2) $x \in E \implies x/n\in E$ $\;$($n=1,2, \;...\;$), (3) $x,y \in E \implies |x-y| \in E$, (4) $x,y \in E \implies ...
1
vote
0answers
67 views

Partition of open sets in $\mathbb{R}^d$.

Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets. In particular we want compact sets $K_j$'s and open sets $V_j$'s such that ...
0
votes
1answer
332 views

Existence of minimum distance between two closed sets, one of which is bounded

I would like to prove the following statement: Let $S,T \subseteq \mathbb{R}^{n}$ be closed sets with $S \cap T = \emptyset$, at least one of which is bounded. Then there exist $x \in S$ and $y \in T$ ...
0
votes
2answers
56 views

Applying a contraction to balls' centers increases the size of the balls' intersection?

The following statement seems clearly true, but I'm having a hard time proving it: Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. For $r\ge 0$, let $B(c,r)\equiv[c-r,c+r]$. Fix ...
4
votes
3answers
144 views

Is concavity of a real-valued function on a Euclidean space implied by concavity of its restriction to every lower dimensional affine subspace?

Consider a function $f$ over $\Re^n$ to $\Re$. Suppose it is true that for every affine subspace with dimension strictly lesser than $n$ the function $f$ is concave. Is the function $f$ concave over ...