1
vote
0answers
98 views

How can I define a measure of similarity between two line segments in $\mathbb{R}^2$?

I have two segments in $\mathbb{R}^2$ and I would like to define a measure of similarity between the two segments. My idea is that I can apply: a scale transformation $s$ in order to equate the ...
2
votes
1answer
23 views

Geometric characterization of an Euclidean norm

Show that $N$ is an Euclidean norm if and only if the intersection of the unit ball with any plane is an ellipse. I'm stuck on this one. I do not see how can I connect the definition of an ...
4
votes
1answer
75 views

A norm on $\mathbb{R}^2$ such that $\partial C$ is the unit sphere?

Suppose we are on $\mathbb{R}^2$. Assume that $C \subset \mathbb{R}^2$ is a convex bounded neighborhood of the origin invariant by central symmetry. Let $\partial C$ denote the boundary of $C$. My ...
1
vote
0answers
70 views

Distance of a point from a subspace vs. diameter

Let X = $(\Bbb R^N, \|\cdot\|)$ be a Banach space. Let $x_0 \in S^{N-1} = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_n^2}=1\}$. Denote $B^N_2 = \{x \in \Bbb R^N : \sqrt{x_1^2+...+x_N^2} \le 1 \}$. Define ...
8
votes
1answer
91 views

For any three vectors $x,y,z\in\mathbb{R}^d$, we have $ \|y-z\|\cdot\|x\|\leq\|x-y\|\cdot\|z\|+\|z-x\|\cdot\|y\|$

Does anyone know a proof of the following problem? Problem: Show that for any three vectors ${\bf x}, {\bf y}, {\bf z}\in \mathbb{R}^d$ the following holds, $$ \|{\bf y} - {\bf z}\|\cdot \|{\bf x}\| ...
0
votes
1answer
73 views

Bounding L2 distance with mean and standard deviation

Let $\mathbf{x}=[x_i]_{i=1}^d, \mathbf{y}=[y_i]_{i=1}^d$ be two vectors in $R^d$. Is it possible to find a lower bound $l\leq \|x-y\|$ and an upper bound $u\geq\|x-y\|$ as a function of ...
5
votes
1answer
124 views

a conjecture on norms and convex functions over polytopes

Suppose one has a convex, bounded polytope P $\subset R^n$ and a strictly convex function $f$ defined everywhere on $R^n$. $f$ has a unique minimum; and suppose this minimum occurs somewhere strictly ...
5
votes
1answer
1k views

Is there a geometric meaning of the Frobenius norm?

I have a positive definite matrix $A$. I am going to choose its Frobenius norm $\|A\|_F^2$ as a cost function and then minimize $\|A\|_F^2$. But I think I need to find a reason to convince people it ...