All Questions
Tagged with convexity polynomials
12 questions
24
votes
4
answers
3k
views
Why the sequence of Bernstein polynomials of $\sqrt x$ is increasing?
Bernstein polynomials preserves nicely several global properties of the function to be approximated: if e.g. $f:[0,1]\to\mathbb R$ is non-negative, or monotone, or convex; or if it has, say, non-...
- 60.5k
11
votes
0
answers
364
views
Is there yet an example of a non-negative convex polynomial that cannot be written as a sum-of-squares?
I have read that it remains an open question, whether an example can be constructed of a non-negative convex polynomial that cannot be written as a sum-of-squares. My reading includes the following ...
- 173
2
votes
0
answers
61
views
Trying to show expected wait is convex -- need to show an expression is positive
I need to show that the following expression is positive
$$ (B+1) (2 B+1) z_0^B-(B+2) (\rho +1) z_0-2 (B+1) (B-1) ((\rho +1) z_0-\rho )+(B-1) (\rho +1) > 0 $$
where $B\geq 1$ is an integer, $0<...
- 63
3
votes
1
answer
667
views
Are polynomials with only real zeros log concave functions?
Consider a polynomial $\sum\limits_{k=0}^n a_kx^k$ with $a_k\geq 0$ and $x\geq 0$.
In this comment, Richard Stanley mentions that polynomials with only real roots are log concave functions. Can ...
- 389
6
votes
0
answers
255
views
Concavity of a function implicitly defined by a polynomial
Consider the following system of $n$ equations:
\begin{equation}f_j^2 = x_j^2\sum_{i=1}^n A_{ij} f_i
\tag{$\star$}
\end{equation}
where $A_{ij}\geq 0$ are known constants and where $x_j>0$ for ...
- 389
5
votes
0
answers
548
views
Log-concave polynomial is a log-concave function?
A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect ...
- 79
2
votes
0
answers
497
views
Given a multivariate polynomial with even degree, can we find its tightest convex polynomial 'envelop'?
To be specific, given a multivariate polynomial function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with even degree $2d$, can we construct a convex polynomial function $g$, such that:
$\forall \mathbf{...
- 81
6
votes
2
answers
718
views
Can we decompose a polynomial into difference of convex polynomials?
Given a multivariate polynomial $p(x_1, ..., x_n)$ on $\mathbb{R}^n$, can we always decompose it into the difference of two convex polynomials? i.e., is there a pair of convex polynomials $f$ and $g$, ...
- 81
7
votes
1
answer
1k
views
Why are all these families of polynomials finally log-concave?
This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$.
(Notation: in the following, the $a_k$ ...
- 13.4k
2
votes
1
answer
188
views
Spline fit with bounded derivations
How can I do a Spline Fit with bounds on some derivations?
Problem
Given:
Set of data points $t_k, x_k$
Set of nodes $n_i$
order $D$ of the spline (in my case $D=5$)
lower and upper bounds $m_d$,$...
- 176
45
votes
4
answers
5k
views
Polynomial roots and convexity
A couple of years ago, I came up with the following question, to which I have no answer to this day. I have asked a few people about this, most of my teachers and some friends, but no one had ever ...
5
votes
5
answers
9k
views
Characterizing convex polynomials
Let $p=\sum_{i=0}^{n}a_ix^i$. Under what conditions on the coefficients $a_i$ is $p$ convex? Strictly convex?
- 51