Let $Q(2)$ be the Banach-Mazur compactum of two-dimensional normed spaces (over the real numbers). Consider the map $\sigma : (0,+\infty) \rightarrow Q(2)$ defined by $\sigma(p) = (\Bbb R ^2,\| \cdot \|_p )$, is $\sigma$ geodesic? I don't even know how to calculate/approximate the distance between $(\Bbb R ^2,\| \cdot \|_p )$ and $(\Bbb R ^2,\| \cdot \|_q )$ for $p$ and $q$ $\ $ "close enough". Maybe I should start with weaker questions like "is this map continuous?".
2026-03-30 08:33:09.1774859589
On the Banach-Mazur compactum, is the map $p \rightarrow (\Bbb R ^2,\| \cdot \|_p )$ a geodesic?
43 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in METRIC-SPACES
- Show that $d:\mathbb{C}\times\mathbb{C}\rightarrow[0,\infty[$ is a metric on $\mathbb{C}$.
- Question on minimizing the infimum distance of a point from a non compact set
- Is hedgehog of countable spininess separable space?
- Lemma 1.8.2 - Convex Bodies: The Brunn-Minkowski Theory
- Closure and Subsets of Normed Vector Spaces
- Is the following set open/closed/compact in the metric space?
- Triangle inequality for metric space where the metric is angles between vectors
- continuous surjective function from $n$-sphere to unit interval
- Show that $f$ with $f(\overline{x})=0$ is continuous for every $\overline{x}\in[0,1]$.
- Help in understanding proof of Heine-Borel Theorem from Simmons
Related Questions in NORMED-SPACES
- How to prove the following equality with matrix norm?
- Closure and Subsets of Normed Vector Spaces
- Exercise 1.105 of Megginson's "An Introduction to Banach Space Theory"
- derive the expectation of exponential function $e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2}$ or its upper bound
- Minimum of the 2-norm
- Show that $\Phi$ is a contraction with a maximum norm.
- Understanding the essential range
- Mean value theorem for functions from $\mathbb R^n \to \mathbb R^n$
- Metric on a linear space is induced by norm if and only if the metric is homogeneous and translation invariant
- Gradient of integral of vector norm
Related Questions in BANACH-SPACES
- Problem 1.70 of Megginson's "An Introduction to Banach Space Theory"
- Is the cartesian product of two Hilbert spaces a Hilbert space?
- Why is $\lambda\mapsto(\lambda\textbf{1}-T)^{-1}$ analytic on $\rho(T)$?
- Is ${C}[0,1],\Bbb{R}$ homeomorphic to any $\Bbb{R^n}$, for an integer $n$?
- Identify $\operatorname{co}(\{e_n:n\in\mathbb N\})$ and $\overline{\operatorname{co}}(\{e_n : n\in\mathbb N\})$ in $c_0$ and $\ell^p$
- Theorem 1.7.9 of Megginson: Completeness is a three-space property.
- A weakly open subset of the unit ball of the Read's space $R$ (an infinite-dimensional Banach space) is unbounded.
- Separability of differentiable functions
- Showing $u_{\lambda}(x):= \left(\frac{\lambda}{{\lambda}^{2}+|x|^2}\right)^{\frac{n-2}{2}}$ is not sequentially compact in $L^{2^{*}}$
- Proving that a composition of bounded operator and trace class operator is trace class
Related Questions in LP-SPACES
- Absolutely continuous functions are dense in $L^1$
- Understanding the essential range
- Problem 1.70 of Megginson's "An Introduction to Banach Space Theory"
- Showing a sequence is in $\ell^1$
- How to conclude that $\ell_\infty$ is not separable from this exercise?
- Calculating the gradient in $L^p$ space when $0<p<1$ and the uderlying set is discrete and finite
- $f_{n} \in L^{p}(X),$ such that $\lVert f_{n}-f_{n+1}\rVert_{p} \leq \frac{1}{n^2}$. Prove $f_{n}$ converges a.e.
- Find a sequence converging in distribution but not weakly
- Elementary use of Hölder inequality
- Identify $\operatorname{co}(\{e_n:n\in\mathbb N\})$ and $\overline{\operatorname{co}}(\{e_n : n\in\mathbb N\})$ in $c_0$ and $\ell^p$
Related Questions in GEODESIC
- Length of geodesic line equals distance between two points?
- What's the relation between the Darboux Frame and the Frenet-Serret on a oriented surface?
- Projection from an ellipsoid onto a sphere that preserves geodesy?
- Vector field on a geodesic
- Geodesic lines of the form f(at+b)
- How to actually find a minimizing path on a manifold?
- Calculating the round metric on $S^n$
- Geodesic equation on a codimension 1 submanifold of $\mathbb{R^{n+1}}$
- How can you numerically approximate the geodesic midpoint of 2 points on an ellipsoids?
- Compute geodesic circles on a Surface
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?