12
votes
Conditions for including cones
Iosif Pinelis already gave a nice solution to show that the answer is "no" for sets of infinitely many vectors, and thus for $N$ very large. I'll show that the answer is also "no" ...
- 6,351
10
votes
Accepted
Closedness of linear image of positive L1 functions
Take $\mathcal X = L^1(\Omega,\mu)$ where $\Omega = \{1,2,3,\dots\}$ and measure $\mu(\{k\}) = p_k$ with $p_k > 0$,
$\sum p_k = 1$. The norm in $\mathcal X$ is
$$
\|f\|_{\mathcal X} = \sum_k |f(k)|...
- 41.1k
8
votes
Conditions for including cones
$\newcommand\R{\mathbb R}$Update: Within the previously established framework, we now show that it is enough to have just $N=4$ points. This improves what seems to be the previous record, of $N=5$. It ...
- 128k
7
votes
Accepted
Image of a quadratic form is a closed cone
In general, $Q(E)$ is not closed.
Counterexample. Let $E = F = \mathbb{R}^2$ and set
$$
B(x,y) :=
\begin{pmatrix}
x_1y_1 \\
x_1y_2 + x_2y_1
\end{pmatrix}.
$$
Then
$$
Q(x) :=
\begin{...
- 12.5k
7
votes
Accepted
Why does every chain complex have a map into its cone?
This is really just rephrasing Simon Henry's comment, but there is a natural transformation $F\to\text{Hom}(C,-)$ given by $(f,s)\mapsto f$, and so if $\text{Cone}(C)$ represents $F$ then by Yoneda's ...
- 35.2k
7
votes
How many cones with angle theta can I pack into the unit sphere?
This is the problem of finding spherical codes. Putatively optimal solutions can be found at Neil Sloane's website.
For an upper bound, there's $d\leq\sqrt{4-csc^2[\frac{πn}{6(n-2)}]}$, where $d$ is ...
- 9,501
5
votes
How many cones with angle theta can I pack into the unit sphere?
A good reference for volumetric arguments for the maximum number of 'cones' or spherical 'caps' that one can fit, is a series of papers by Jon Hamkins. The density of a packing of these caps can be ...
- 3,209
4
votes
Accepted
Closed convex cone - equivalence of definition via closure and via infinite sums
The answer to your question is "No" (but the first set, obviously, always contains the second).
Example showing that the second set can be strictly smaller: Denote by $\{e_n\}$ the unit vector basis ...
- 902
4
votes
Accepted
A characterisation of faces of rational polyhedral cones
To prove that (iii) implies (i), assume w.l.o.g. that $\tau\neq\sigma$. We first need to show that $\tau\subset \partial \sigma$. If this is not the case then either there exists a hyperplane $H$ ...
- 14k
4
votes
Semisimple Lie algebra and convexity
I'm not super familiar with the theory of Jordan algebras or self-dual homogeneous cones so I might be barking up the wrong tree with this answer but here goes. I don't think the cones I'm about to ...
- 954
4
votes
Accepted
Subdifferential of a convex function admits a continuous selection
$\newcommand\p\partial\newcommand\R{\mathbb R}\newcommand\cl{\operatorname{cl}}\newcommand\conv{\operatorname{conv}}$The answer is yes, and you were almost there.
Indeed, suppose the contrary: that we ...
- 128k
3
votes
Accepted
Interpolation of normed spaces *vs* geometrical mean of positive matrices
Yes. The proof of Theorem 1.1 from John E. McCarthy's "Geometric interpolation between Hilbert spaces," Ark. Mat. 30, 321-330 (1991) works for this case. Let $A_i$, $B_i$, be SPD matrices, $...
- 621
3
votes
Accepted
Finding Motzkin's original paper on copositive quadratic forms
Hall and Newman (1963) cite this work as
Motzkin, T., Copositive quadratic forms. National Bureau of Standards Report 1818 (1952), pp. 11–12.
This cited part of the NBS report is available online at ...
- 34.3k
3
votes
Separation of two pointed polyhedral cones using hyperplanes generated by facets
Yes for dimension 2 (pick the cone with larger angle; one of its sides fits). No for larger dimensions.
For a counterexample in dimension $3$, let $C_1$ be a positive orthant, and let $C_2$ be a ...
- 23.7k
2
votes
Accepted
1st Order Nonlinear PDE: Understanding Envelopes and Monge Cones
A geometric interpretation of the envelope of a one-parameter family of surfaces is the following (slightly adapted from Wikipedia): a point on the envelope is a point in the intersection of two "...
- 5,343
2
votes
Accepted
Projection onto the second-order cone
The proof can be found in H.H. Bauschke's 1996 doctoral dissertation: Projection Algorithms and Monotone Operators (p. 40, Theorem 3.3.6).
P.S. I wonder what the downvote is for.
- 1,538
2
votes
Cone construction for Birkhoff Hopf theorem
A positive matrix is primitive and so it has only one nonzero eigenvalue of maximum modulus.
An irreducible which is not primitive has (by definition) strictly more than one eigenvalue with maximum ...
- 682
2
votes
Accepted
Dual cone of 'positive' Bochner integrable functions
Take some element in the dual cone $g\in L^1([0,1],Y_+)^*\subset L^\infty_{w^*}([0,1],Y^*)$.
Then, by definition, for every $f\in L^1([0,1],Y_+)$
$$\int_0^1 \langle f(t),g(t)\rangle\mathbb{d}t\geq0$$...
- 138
2
votes
Accepted
Intersection of a closed convex cone with the non-negative orthant
Assume $C$ and $\mathbb{R}_{\ge 0}^n$ can be (non-strictly) separated by a subspace of dimension $n-1$.
Then a normal vector $x$ to that subspace lies in $\mathbb{R}_{\ge 0}^n$; see e.g. here for a ...
- 233
2
votes
Order bounded version of monotone complete $C^*$-algebras
Let $\tilde{A}$ denote the (unique up to isomorphism) unitization of a non-unital $C^*$-algebra $A$.
Proposition 2.1.11 in Saitô and Wright's monograph states the following:
Let $A$ be a $C^*$-...
- 21
2
votes
Accepted
Do highly symmetric cones have "small" supporting hyperplanes?
The prescribed group action allows to start from it and build the cones in question. It is well-known how the permutation representation of $S_n$ on 3-subsets decomposes into irreducibles - there will ...
- 14k
1
vote
Are the automorphism groups of simple symmetric cones algebraic groups?
I think I figure a proof after discussing with Yu Zhao.
First of all, by Koecher-Vinberg theorem, we can put a structure of Euclidean Jordan algebra on $V$ so that the closure $\bar{C}$ of $C$ is the ...
- 329
1
vote
What is the convex cone generated by the pair of rank 1 matrix and its eigenvector?
Are you really interested in the convex cone, or in the convex envelop ? The latter is easily determined by taking the intersection of the half-spaces containing all the pairs $(hh^T,h)$. Their ...
- 52.3k
1
vote
Accepted
Affine cone example
The algebra that you are describing gives the fibre over $X$ in the blow-up of $\mathbb{P}^n$ along $X$. (The blow-up is given by $\mathrm{Proj}(\bigoplus_{n=0}^\infty I^n$.) Therefore, in order to ...
- 661
1
vote
Faces of polyhedral cones and open immersions of affine toric schemes
For $R=\mathbb{C}$, this is exercise 3.2.10 in Cox-Little-Schenck (most likely for the proof outlined there this assumption is not really necessary).
- 1,030
1
vote
Sensitivity analysis in conic optimization
This bibliography should help:
"Semidefinite and Cone Programming Bibliography/Comments", Henry Wolkowicz, 2011 (http://www.math.uwaterloo.ca/~hwolkowi/henry/software/sdpbibliog.pdf)
...or have you ...
- 11
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