# Questions tagged [norms]

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### Is there a relationship between infinity norm (or any other norms) of a vector and the trace of its covariance matrix?

I wish to know if there is a known relationship between the infinity norm (or any other norms) of a vector and the trace of its covariance matrix? I have found a paper that used the following ...
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### Regularity of functions everywhere approximable by $n$-th degree polynomials

Let $(X, \lVert \cdot \rVert_X)$, $(Y, \lVert \cdot \rVert_Y)$ be two Banach spaces. A function $P \colon X \to Y$ such that there exists $n \in \mathbb{N}$ such that for all $i \in \{ 0, \ldots, n \}$...
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### Probability of $\ell_1$-norms of vertices of the rotated Hamming cube

Let $O$ be a $d$-dimensional rotation matrix (i.e., it has real entries and $OO^T = O^TO = I$). Let $\mathbf{x}$ be a uniformly random bitstring of length $d$, i.e., $\mathbf{x} \sim U(\{0,1\}^d)$. In ...
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### A generalized norm function in $\mathbb{R}^n$ [closed]

We defined a new norm. The norm of $x \in \mathbb{R}^n$ is defined as $$N_P(x) = \min \{t \geq 0 : x \in t\cdot P\} \enspace,$$ where $P$ is a centrally symmetric and convex body centered at the ...
1 vote
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### Double space integral formulation of homogeneous Sobolev norm

Define for $u\in C_c^\infty (\mathbb R^n), 0<s<1$ the integral $$I_s(u) = \int_{(x,y)\in \mathbb R^{n+n}} \frac{(u(x+y)-u(x))^2}{|y|^{d+2s}} dxdy.$$ I wish to prove that for some $C=C(s)>1,$...
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### Weighted metric (or semi-metric) with incorporated distance between Dimensions

Im trying to construct a distance Measure between two vectors, that takes into account also the distance between the Dimensions. I will illustrate my Problem with some examples: $x,y \in \mathbb{R}^n$,...
I've previously asked this question on stack exchange. I'm looking for a proof of the inequality $$\left[ \frac12(\left\|A+B\right\|_p^p + \left\|A-B\right\|_p^p)\right]^{2/p} \leq \left\|A\right\|_p^... 3 votes 1 answer 57 views ### Estimate of the norm of the radial part of a function Consider a function u\in L^2(\mathbb R^N), and another function \varphi which is the unique solution to the Poisson equation \Delta \varphi = u vanishing at \infty. We know that the radial ... 1 vote 1 answer 104 views ### Norm of a matrix with clustered eigenvalues On page 271 of Trefethen and Bau's Numerical Linear Algebra, it is constructed a matrix$$A=2I_{m\times m}+0.5\cdot\frac{\text{rand}(m)}{\sqrt{m}}$$for m=200, where rand(m) is an array with m\... 7 votes 3 answers 1k views ### Bounding supremum norm of Lipschitz function by L1 norm Consider f:[0,1]^d \to \mathbb{R}. Suppose that f is L-Lipschitz w.r.t. the Euclidean norm. Can we provide an upper bound on \|f\|_\infty in terms of \|f\|_1 := \int_{[0,1]^d} |f(x)|dx ? In ... 12 votes 1 answer 402 views ### Subtracting the weak limit reduces the norm in the limit Question Let X be some reflexive Banach space. Suppose x_n is some sequence in X that weak converges to some y \neq 0. Is it the case that$$ \limsup \|x_n - y\| < \limsup \|x_n\| ? ...
This is probably known, but I have not located a reference. Let $P$ be the convex hull of $k$ points in $\mathbb R^n$ with rational coordinates. Consider the Euclidean square norm function \$F:P\to\...