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20

So it turns out my earlier intuition was incorrect, and one can leverage the order properties of ${\bf R}$ to show: Theorem. Let $X, Y$ be real Hilbert spaces, with the dimension of $X$ at least two, and let $f: X \to Y$ be a function such that $\|f(x)-f(y)\|=\|x-y\|$ whenever $\|x-y\|$ is a natural number. (We do not assume $f$ to be continuous.) Then ...


19

Well, as you have certainly already remarked (reading your post, I assume this), bilinearity makes a big difference. For the "only norm" case, what you are looking for, if I understand correctly your question, is a set of uniqueness for the admissible norms on a given vector space $V$. Your demonstration establishes that values on a dense set $U$ (on the ...


18

If $E$ is to be a Hilbert space, a proof must establish more or less directly that the inequality implies the parallelogram law $\lVert x + y\rVert^2 + \lVert x-y\rVert^2 = 2\lVert x\rVert^2 + 2\lVert y\rVert^2$ for all $x,y \in E$. Since both, your hypothesis and the parallelogram law, are conditions on all $2$-dimensional subspaces of $E$, one can assume ...


13

Let $X=Y=\mathbb R$ with the absolute value norm and define $n(a,b)=\sqrt{2a^2+2b^2-3ab}$. This is a norm on $\mathbb R^2$ because it is the quadratic form of the positive definite matrix $A=\left( \begin{smallmatrix} 2 & -3/2 \\ -3/2 & 2 \end{smallmatrix}\right)$. Then $N(v) = n(|v_1|,|v_2|)$ is not a norm because the triangle inequality fails for ...


13

This is only an illustration to Christian Remling's beautiful answer; here are the concentric "balls" around the origin for this "norm":


13

Metric space $(X,\rho)$ satisfying Ptolemy inequality $\rho(a,b)\rho(c,d)+\rho(b,c)\rho(a,d)\geq \rho(a,c)\rho(b,d)$ is called ptolemaic space. A normed ptolemaic space must be inner product space. Reference: I.J. Schoenberg, A remark on M. M. Day’s characterization of innerproduct spaces and a conjecture of L. M. Blumenthal. Proc. Am. Math. Soc. 3, 961–964 ...


10

In many books$^*$ you can find the result that there is a projection of norm at most $\sqrt{n}$ onto any $n$ dimensional subspace of a Banach space. For reflexive spaces, this gives immediately that every $n$ codimensional subspace is the range of a projection that has norm at most $\sqrt{n} +1$. For non reflexive spaces, by using the principle of local ...


10

If you assume $f$ to be surjective then $f$ has to be linear without any assumptions on $V$ by the Mazur-Ulam theorem. Wikipedia doesn't offer much more information than a link to the beautiful recent proof by J. Väisälä.


8

You might be interested in the whole (mostly Russian) literature on "Banach-Kantorovich" or "lattice-normed" spaces, which are: "a triplet $(\mathcal U,E,\lambda)$ consisting of a vector space $\mathcal U$, a Dedekind complete vector lattice $E$ and a map $\lambda:\mathcal U\to E_+$ satisfying some natural conditions that allow one to consider $\lambda$ ...


8

The closest thing I know for induced norms is the Riesz–Thorin theorem. There are other Hölder-like inequalities for matrices, for example involving Schatten norms.


8

Terry Tao's argument generalizes to show that $f$ must be an isometry whenever $X$ has dimension greater than 1 and $Y$ is strictly convex. We start by proving a series of lemmas: Lemma 1: Let $x,y\in X$, and let $a,b\geq0$ be such that $a+b\geq\|x-y\|$ and $|a-b|\leq\|x-y\|$. Then there exists $z\in X$ such that $\|x-z\|=a$ and $\|y-z\|=b$. Proof: Let ...


7

This is a comment rather than an answer but I am not entitled. The missing requirement on the two dimensional norm for the statement to be valid is that it be increasing in the natural sense that if $|x|\leq|x_1|, |y|\leq|y_1|$, then $n(x,y)\leq n(x_1,y_1)$. Norms without this property can easily be obtained by rotating a non-circular ellipse in standard ...


7

$\def\sign{\mathop{\rm sign}}$First of all, it is enough to prove the statement when $p=u/v$ is rational, $u$ is even and $v$ is odd (such numbers are dense on the real line). We need this to simplify the last argument. Let the equation of the ellipse be $f(x,y)=ax^2+2bxy+cy^2=1$; one may assume that $b\neq 0$ (either by a small variation argument, or by ...


7

I got $$r([A,B])\leq 4\sqrt{2} r(A) r(B).$$ It is lower than $8$ but still higher than the conjuncture $C_{nr}=4$. I used the following facts: For normal (i.e. $X^\*X=XX^\*$) matrices we have $r(X)=\sigma_1(X)$ (the largest singular value of X). $\sigma_1(XY-YX)\leq 2 \sigma_1(X)\sigma_1(Y)$ Also, note that for $X$ and $Y$ hermitian (i.e. $X^\*= X$ ...


6

this is called the numerical radius of a matrix


6

By the Courant-Fischer min-max theorem, if $A$ is Hermitian, then $f(A) = \lambda_n(A)$, the smallest eigenvalue of $A$.


6

Take a look at the papers "Über Normtopologien in linearen Räumen" (Link to the article) as well as "Über vollständige Normtopologien in linearen Räumen" (Link to the article) by D. Laugwitz. He proves the following. Let $E$ be a vector space, denote by $a(E)$ the cardinality of a Hamel basis of $E$ and denote by $n(E)$ the number of mutually non-equivalent ...


5

(This answer expands the ideas in the comments by Mikael de la Salle and myself). There is a family of matrix norms that can be defined as follows: given $p \geq 1,k \leq n$, $$ \|A\|_{p,k} = \left(\sum_{i=1}^k \sigma_i(A)^p\right)^{1/p}, $$ where we denote by $\sigma_i(A)$ the $i$th singular value of $A$ (taken in nonincreasing order, $\sigma_1(A)\geq ...


5

$|||A|||$ is the largest s-number (modulus of gen. eigenvalue). $f(A)$ is the smallest s-number. It is 0 if $A$ is not injective.


5

Here is a partial result. Claim. $4 \le C_{nr} \le 8$. Proof. The upper bound has already been shown by the OP. The lower-bound follows by \begin{equation*} A = \begin{bmatrix} 0 & 1\\\\ 0 & 0 \end{bmatrix},\qquad B = \begin{bmatrix} 0 & 0\\\\ -1 & 0 \end{bmatrix} \end{equation*} for which ...


5

The answer by Piotr Migdal can be modified to give the accurate inequality $$r([A,B])\le4r(A)r(B),\qquad\forall A,B\in{\bf M}_n(\mathbb C).$$ The only new argument is that for every matrix $M$, there exists an angle $\theta$ such that $r(M)=\|{\rm Re}(e^{-i\theta}M)\|_2.$ Actually, we do have $$r(M)=\sup_\alpha\|{\rm Re}(e^{-i\alpha}M)\|_2.$$ Hereabove, ...


5

The best such inequality that depends only on $m$ and $n$ is: $$ \frac{1}{\sqrt{mn}}\|A\| \leq \|A\|_* \leq \|A\| $$ The right inequality is tight when $A$ is a matrix with a $1$ in the top-left corner and zeroes elsewhere. The left inequality is tight when $A$ is the matrix all of whose entries are $1$. These examples also show that you cannot get any ...


4

The following argument seems easier, but there might be a still more fundamental one. Notice that $ \phi: A^{\sim} \to \mathbb{C} $ above is also a $ C^{*} $-algebraic homomorphism. As $ C^{*} $-algebraic homomorphisms are automatically contractive (which is a consequence of a not-too-difficult spectrum argument), we have $$ \forall (a,\lambda) \in A \times ...


4

Here is a simple proof that the property holds only for Euclidean norms, at least if the norm in question is $C^1$ smooth and strictly convex. Surely it was known way before Gromov was born. Let $S$ denote the unit sphere of the norm. First observe that, if $v\in S$ and $w$ are such that $\|w+tv\|\ge \|tv\|$ for all $t\in\mathbb R$, than $w$ is parallel to ...


4

You can use the surprising identity $(A^{-1}+B^{-1})^{-1}=A(A+B)^{-1}B$, and take the norms of the three factors separately.


4

Yes, a classical result says that every 2-dimensional (real) normed space embeds into L^1. Alternatively, if we assume that the unit ball is a $2n$-gon (the general case then follows by approximation), the corresponding space embeds into $\ell_1^n$. The dual picture is maybe even more transparent : any symmetric $2n$-gon is the Minkoswki sum of $n$ ...


4

Computing such induced norms is a hard problem. For the case of $p=2$ and $q \ge 2$, have a look at this paper by Barak et al. to see how tricky the problem is. Typically, for other than the nice cases of $1,2, \infty$ style, these norms are NP-hard to compute, with well-known results for the case $p \ge q$ (see also e.g.: "Matrix norms are NP-Hard to ...


4

The question that you are asking was asked in "On Generalizations of Minkowski's Inequality in the Form of a Triangle Inequality" by F. Mulholland (1949). In that paper, Mulholland established a sufficient condition on $f$, namely that it should satisfy $f(0)=0$, be increasing on $x \ge 0$ and be g-convex, i.e., $\log f(e^x)$ is convex on the reals. ...


4

(This isn't an answer, but I'm not allowed to comment yet and I think this is a worthwhile question, even if it means violating the rules; i'll delete the post when the time comes) You claim that usual matrix norms do not work because of the $|f(x_1)-f(x_2)|$ criterion, but if $f$ is differentiable, then given any $x_1$ it is Lipschitz in some neighborhood ...


4

Maybe I am misinterpreting something, because according to my experiments, this function is neither convex nor concave. The following is a counterexample (EDIT: I changed the example to use symmetric matrices): \begin{equation*} A=\begin{pmatrix}8 &4\\ 4 & 6\end{pmatrix},\quad B=\begin{pmatrix} 4 & 4\\ 4 & 6\end{pmatrix},\quad ...



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