21
votes
Accepted
Prove that this expression is greater than 1/2
Let
$$f(x,y):=4x^{2}+4y^{2}-4xy-4y+1 + \frac{4}{\pi^2}\Bigl(
\sin^{2}(\pi x)+ \sin^{2}(\pi y) + \sin^{2}(\pi y-\pi x) \Bigr).$$
I will show that
$$\min_{0\leq x\leq y\leq 1}f(x,y)=\min_{0\leq x\leq 1/...
- 99.1k
19
votes
Accepted
Illustrating that universal optimality is stronger than sphere packing
In three dimensions you don’t need to go beyond lattices to see the failure of universal optimality. When the potential function is sufficiently steep (e.g., a narrow Gaussian), the face-centered ...
- 16.6k
11
votes
"Small" maps from sphere to sphere
Here's an example to show that the infimum is not always attained:
Consider the standard Hopf map $\pi:S^3\to S^2$, which is not null-homotopic, of course, so it follows that the area of the graph in ...
- 106k
11
votes
Accepted
Known configurations maximizing the volume of the convex hull of n points on the unit sphere
The problem is elementary for $n=5$.
We may regard that case as a combination of two triangular pyramids sharing a base $\triangle ABC$. Then the volume is always bounded by one-third the area of the ...
- 1,603
11
votes
Accepted
Min problem on integers
Let us denote
$$\sigma_\ell:=\sum_{i=1}^\ell q_i\qquad\text{and}\qquad\tau_\ell:=\sum_{i=\ell+1}^s\frac{1}{q_i}.$$
Then
$$\prod_{\ell=1}^{s-1}\left(\frac{q_\ell}{q_{\ell+1}}\cdot\frac{\sigma_{\ell+1}}{...
- 99.1k
10
votes
Prove that this expression is greater than 1/2
Let $F(x,y)$ denote the left-hand side of your inequality. It is easy to see that $|\nabla F(x,y)|\sqrt2/n<0.002$ if $0<x<y<1$, where $n:=6600$. A direct calculation shows that $F(i/n,j/n)&...
- 118k
7
votes
Accepted
Maximal distance between $2d+1$ points on the $(d-1)$-sphere
I finally came to do the computations on Yoav Kallus' comment. After quite some tedious work one finds a cubic polynomial:
$$p_d(x)\,=\,d(d-2)^2 x^3 - d^2 x^2 - dx + 1$$
which has exactly one zero in ...
- 12.6k
6
votes
Prove that this expression is greater than 1/2
$\newcommand{\R}{\mathbb{R}}$
An advantage of my previous answer was that, while the computer calculations were pretty heavy there, the logic was extremely simple; virtually no thinking or ingenuity ...
- 118k
6
votes
Maximal distance between $2d+1$ points on the $(d-1)$-sphere
OK, it's late and I may be wrong but I think that you can obtain the $2d$ points by using any set of orthonormal basis vectors $\{v_1,\ldots,v_d\}$ and their negatives.
Now if $n$ is such that a ...
- 10.1k
6
votes
Maximal distance between $2d+1$ points on the $(d-1)$-sphere
Turns out kodlu's idea works in all dimensions, regardless of the existence of any Hadamard matrices.
Consider all coordinate permutations of
$$(1,...,1,-d)\in\Bbb R^{d+1}\quad\text{and}\quad (-1,......
- 12.6k
5
votes
Solve $\inf_{ X: |X| \le a \text{ a.s.}} E \left[ \frac{1}{1+(X-X^\prime)^2} \right] $
My initial intuition was incorrect. It seems we can sometimes solve this explicitly, and the minimizer does not have full support; my argument is incomplete (as discussed below), but I think it's ...
5
votes
Accepted
A (reverse)-Minkowski type inequality for symmetric sums
Rewrite the inequality in question as
\begin{equation*}
f(u+v)\le f(u)+f(v)
\end{equation*}
for $u,v$ in $\mathbb R_+^4$,
where
\begin{equation*}
f(u):=-\left(\left(\frac{1}{\sqrt{u_1}}+\frac{1}{...
- 118k
5
votes
Does there exist energy-minimizing immersions?
I believe that in general, without any additional assumptions about the manifolds, the answer is in the negative. A counterexample can be found for mappings between annuli. Let $A=A(r,R)$ and $A_*=A(...
- 27.3k
5
votes
Accepted
Is this lower bound for a norm of some complex matrices true?
No, it is not; in fact, $2(n-1)$ is a local maximum.
Let $B$ be a Hermitian matrix such that $|B_{ij}|=1$ and $B_{ii}=1$. We denote its eigenvalues by $\mu$ (not to confuse them with eigenvalues of $...
- 6,861
5
votes
Can solutions to Thomson's problem have pentagons?
Technically, there are best known configurations with pentagons and even with hexagons -- if we allow these polygons to loop around multiple faces. Such configurations may be read from the Wikipedia ...
- 1,603
4
votes
Accepted
Differences between the convex discrete maximization and minimization problems?
For continuous problems, minimizing a convex function on a convex domain is considered an easy problem, because there is only ever one local minimum, and a local minimum is the global minimum. ...
- 53.6k
4
votes
Can you give me good examples of non-convex functions that are problematic for optimization?
A less-known example is $f(x):=x^2+\exp(-1/(100(x-1))^2)-1$ on the closed interval $[-2,2]$. It takes $-.0067419337989203 $ at $x = .996387676055289 $.
See that discussion in MaplePrimes for more ...
- 3,309
4
votes
A (reverse)-Minkowski type inequality for symmetric sums
The said claim follows from the following general result on elementary symmetric polynomials, denoted $e_k$ below.
$\newcommand{\vx}{\mathbf{x}}\newcommand{\vy}{\mathbf{y}}$
Theorem A (S. 2018). $\,...
- 28.4k
4
votes
Accepted
Maximizing the $\alpha$-moment of a distributution
Let us consider the closely related problem: maximize $EX^\alpha$ over all nonnegative random variables (r.v.'s) $X$ with $EX=\mu$. To avoid trivialities, assume that $\mu\in(0,\infty)$. Consider the ...
- 118k
4
votes
Accepted
Maximizing a convex function with a convex constraint
Under your assumptions, this is a concave programming problem (i.e., minimization of a concave function subject to convex constraints) with compact constraint set, and therefore has a global minimum ...
- 1,462
4
votes
Accepted
Measurable selection for argmin to distance
Here is a simple direct argument. For $i=1,\ldots,n$, let $$C_i=\{z\in Z\mid d(z,x_i)\leq d(z,x_j), j=1,\ldots,n\}.$$
Clearly, each $C_i$ is closed and hence measurable. Let $M_i=C_i\setminus\bigcup_{...
- 12.6k
4
votes
optimization over moving domains
$\newcommand\R{\mathbb R}$The answer is no.
E.g., let $A=B=\R$, $B_a=[1-a^2,2]$ for $a\in A$, and $\ell(b)=b^3-3b$ for $b\in B$. Then $\ell(b)$ and $B_a$ are perfectly smooth, but $L$ is not ...
- 118k
3
votes
Solve $\inf_{ X: |X| \le a \text{ a.s.}} E \left[ \frac{1}{1+(X-X^\prime)^2} \right] $
You ask what is the minimiser of the functional
$$ F(\mu) = \int_{[-a,a]} \int_{[-a,a]} \frac{1}{1+(x-y)^2} \mu(dx) \mu(dy) . $$
As Christian Remling points out, it is unlikely a closed-form ...
- 16.3k
3
votes
Accepted
minimum of convex function in different variables
This is false. Here is a graph of $\min(x^2, y^2+7).$
- 95.6k
3
votes
Maximizing a pseudoconcave function in a box
Your problem is a special case of a Fractional Linear Program, so as such following the recipe provided on Wikipedia you should be able to solve it by using a reformulation to an equivalent linear ...
- 28.4k
3
votes
Is this lower bound for a norm of some complex matrices true?
This is merely to expand on one of my comments above.
$\newcommand{\Tr}{{\rm Tr}}$
$\newcommand{\snorm}[2]{\Vert#2\Vert_{(#1)}}$
We recall that for any complex $n\times n$ matrix $X$, the trace norm ...
- 25.5k
3
votes
Optimization problem on trace with both the positive semi definite and non positive semidefinite matrix
$\DeclareMathOperator{tr}{tr}$
$\DeclareMathOperator{grad}{grad}$
$\DeclareMathOperator{sym}{sym}$
New answer:
WLOG, We can assume that $B$ is positive definite. Since for every orthogonal matrix $U$...
- 1,991
3
votes
Optimization problem on trace with both the positive semi definite and non positive semidefinite matrix
For Hermitian matrices $A$ and $B$, we know that the following inequality holds:
\begin{equation*}
\text{tr}(AB) \le \langle \lambda^{\downarrow}(A), \lambda^{\downarrow}(B)\rangle,
\end{equation*}
...
- 28.4k
3
votes
Prove that this expression is greater than 1/2
By the Max Alekseyev's hint we need to prove that $\sum\limits_{cyc}f(a)\geq\frac{3}{4},$
where $f(x)=x^2+\frac{2}{\pi^2}\sin^2\pi x,$ $a$, $b$ and $c$ are positives such that $a+b+c=1.$
We have $$...
- 1,054
3
votes
Accepted
Does the plane clustered to minimize sum distances^2 to clusters centers ( inertia / "K-means") produce hexagonal clusters / hexagonal lattice?
The answer is yes, at least in the limiting case where the number of points tends to infinity.
Specifically, this is known as the quantizer problem (see Chapter 2 of Sphere Packings, Lattices and ...
- 12.2k
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