Intereting Posts

How to tell $i$ from $-i$?
Proof that $\mathbb{Z}\left$ is a PID
Is $(XY – 1)$ a maximal ideal in $k]$?
Counting the Number of Integral Solutions to $x^2+dy^2 = n$
Antipodal mapping of the sphere
Conditional expectation $E$ for $X$ and $Y$ independent and normal
References on filter quantifiers
equivalence between uniform and normal distribution
How many total order relations on a set $A$?
Can non-constant functions have the IVP and have local extremum everywhere?
Solving the differential equation $\frac{dy}{dx}=\frac{3x+4y+7}{x-2y-11}$
A question about the equivalence relation on the localization of a ring.
What is the best base to use?
number of non-isomorphic rings of order $135$
Does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$?

Is a uniquely geodesic space contractible ?

We assume in addition that closed metric balls are compact.

A post without this extra assumption is here.

- Fixed point of interior of closed disk
- invariance of integrals for homotopy equivalent spaces
- Which manifolds are parallelizable?
- Can two different topological spaces cover each other?
- Is the classifying space $B^nG$ the Eilenberg-MacLane space $K(G, n)$?
- Homology groups of torus

- seemingly nontrivial question about covering maps and evenly covered open sets
- If $f$ is continuous then $G$ is homeomorphic to $X$.
- Dual space $E'$ is metrizable iff $E$ has a countable basis
- Is a norm a continuous function?
- Are there uncountably many non homeomorphic ways to topologize a countably infinite set?
- Show $\Omega$ is simply connected if every harmonic function has a conjugate
- A question on a compact space
- Is this proof for Theorem 16.4 Munkers Topology correct?
- Does every homeomorphism of a compact metric space lift to the Cantor set?
- $f:\mathbb{S}^1\rightarrow\mathbb{S}^1$ odd $\Rightarrow$ $\mathrm{deg}(f)$ odd (Borsuk-Ulam theorem)

Let us assume in addition that closed metric balls in your metric space $X$ are compact (I do not know what to say in general). I will use the notation $g(p,q)$ for the unique geodesic in $X$ connecting $p$ to $q$ and regarded as a map from

an interval $[0,D]$ to $X$, where $D$ is the distance from $p$ to $q$.

Pick a point $x_0\in X$ and define a family of maps $f_s: X\to X$ by $f_s(x)$ defined to be the point on the geodesic $g(x_0, x)$ so that the distance from $x_0$ to $f_s(x)$ is $s$ times the distance from $x_0$ to $x$. Clearly, $f_1(x)=x$, $f_0(x)=x_0$ for all $x\in X$. What is not clear is if the map is continuous. I think, it would follow from continuity of the map sending $x$ to the geodesic $g(x_0, x)$ (where I use topology of uniform convergence on the set of geodesics). Suppose that I have a sequence $x_n\in X$ convergent to $x\in X$. I claim that such sequence always contains a subsequence so that the corresponding sequence of geodesics converges to $g$.

By the compactness assumption I made and the Arzela-Ascoli theorem, the sequence of geodesics $g_n$ (regarded as maps of course) converges (after passing to a subsequence) to a geodesic $g’$ connecting $x_0$ to $x$. (Usually the A-A theorem is applied to maps with a fixed domain. In our case, the sequence of domains is a sequence of intervals $[0, D_n]$ where $D_n$ is the distance form $x_0$ to $x_n$. To resolve this issue, take some $D\ge D_n, \forall n,$ and extend the maps $g_n$ to $[0,D]$ by constant maps on intervals $[D_n,D]$.)

Since $X$ is uniquely geodesic, $g’=g$. Hence, our subsequence in $g_n$ converges to $g$. Now, use the fact that in a compact metric space $G$ (the space of suitable geodesics in our case), a sequence $g_n$ converges to $g$ if and only if every convergent subsequence in $(g_n)$ converges to $g$. Thus, we proved that the sequence of geodesics $(g_n)$ converges to $g$ and, hence, the map from $X$ to the space of geodesics is continuous.

- Find $\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+…}}}}$
- If $x_1 = 3$, $x_{n+1} = \frac{1}{4-x_n}$ for $n \geq 1$, prove the sequence is bounded below by $0$, above by $4$.
- Proving that a convex function is Lipschitz
- Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$
- What are the zero divisors of $C$?
- Local property of real submanifold embedded in complex space
- $Z(t)=Xt+Y$ is a random process find the following
- Continuous Collatz Conjecture
- $\int_0^\infty ne^{-nx}\sin\left(\frac1{x}\right)\;dx\to ?$ as $n\to\infty$
- $2^a +1$ is not divisible by $2^b-1$.
- Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$
- Isomorphism in localization (tensor product)
- characteristic polynomial of companion matrix
- What makes Tarski Grothendieck set theory non-empty?
- How to construct this Laurent series?