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2
votes
1answer
77 views

small perturbation of BV function

consider an interesting real analysis question: define average operator on $[0,1]$: $A_{\epsilon} f (x) = \frac{1}{2\epsilon}\int_{x-\epsilon}^{x+\epsilon} f(y) dy , f \in BV[0,1] $ ( may clarify ...
4
votes
1answer
113 views

Is the preimage of a face under an affine map a face?

Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $\pi{:}\ X\to\mathbb{R}^m$ be a linear map, with $m<n$ (for example, a projection). Let $\pi^{-1}(y)=\{x\in X\mid\pi(x)=y\}$ denote the inverse of $...
2
votes
0answers
43 views

Relative interior of subdifferential

$\def\ri{\mathop{\rm ri}}$ Conjecture: Let $f{:}\ \mathbb{R}^n\to\mathbb{R}$ be convex and let $x,y\in\mathbb{R}^n$. If $0\in\ri\partial f(x)\cap\ri\partial f(y)$ then $\partial f(x)=\partial f(y)$. ...
4
votes
2answers
117 views

A kind of exponential concavity for polynomials?

Is there $C > 0$ such that the inequality $$ \prod_{n\in\mathbb{N}} p(n)^{a_n} \leq p\left(C\prod_{n\in\mathbb{N}} n^{a_n}\right) $$ holds for all finitely supported sequences $(a_n)$ with $a_n\geq ...
9
votes
1answer
247 views

Is KL divergence $D(P||Q)$ strongly convex over $P$ in infinite dimension

By KL divergence I mean $D(P||Q) = \int dP \log(\frac{dP}{dQ})$. I am looking for the conditions under which this strong convexity is true and possible references. I could not find an answer for ...
3
votes
1answer
185 views

concavity of $\log [ (1+\frac{x_0}{1+x_0+x_1/2+x_2/4}) (1+\frac{x_1}{1+x_0/4+x_1+x_2/2}) (1+\frac{x_2}{1+x_0/2+x_1/4+x_2}) ]$

Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be $$ f(x_0,x_1,x_2)= \log \left[ \left(1+\frac{x_0}{1+x_0+\frac{x_1}{2}+\frac{x_2}{4}}\right) \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_0}{4}}\right) \...
1
vote
0answers
43 views

Star-convex curve and Fourier series

Let x(t) be a periodic function on [0, 2$\pi$]. I am interested in finding criteria in terms of the Fourier coefficients of x(t), such that the parametric curve $\left\{ x\left( t\right) ,\dot{x}\left(...
4
votes
3answers
138 views

Quasi-concavity of $f(x)=\frac{1}{x+1} \int_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt$

I want to prove that function \begin{equation} f(x)=\frac{1}{x+1} \int\limits_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt \end{equation} is quasi-concave. One approach is to obtain the closed form ...
1
vote
1answer
73 views

Every closed and convex subset of a uniformly convex metric space is Chebyshev?

I came across the statement ``Every closed and convex subset of a uniformly convex b-metric space is Chebyshev'' in [1]. Here, the term `convex' is in the sense of Takahashi. I tried looking up for ...
1
vote
0answers
49 views

Are the sets whose convex hull surface admits multiple representations a shy set of sets?

Consider a compact subset $A$ of $R^n$. Let me call $A$ special if any point $x$ that is on the boundary of $\textsf{conv.hull}(A)$ admits a unique representation as a convex combination of points in $...
0
votes
0answers
69 views

Volume of parametric integral of convex set

Given $t,\mu>0$. I am interested in computing the volume of the $n$-dimesnional set $$\int_{0}^{t}e^{A(t-\tau)}\begin{pmatrix}0\\ 0\\ \vdots\\ 0\\1\end{pmatrix}U\:{\rm{d}}\tau,$$ where the set $U$ ...
2
votes
1answer
108 views

Quasi-concavity of $f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$ on $[0~1000]$

I want to prove that function $f:[0~1000]\rightarrow R$, $$f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$$ is quasi-concave. Any idea how to do the proof? I already tried to prove that any super-level set is ...
4
votes
2answers
129 views

Bound on sum of $n$th super-diagonal entries in a $2n$ by $2n$ PSD matrix

Let $A,B,C\in\mathbb{R}^{n\times n}$ be such that $\left(\begin{array}{} A & B \\ B^T & C \end{array}\right)\succeq 0$. I would like to prove that $$\mathrm{trace}\,B \le \sum_{i=1}^n \sqrt{\...
0
votes
0answers
44 views

Optimization of generalized Fisher information over space of probability measures

Assume we already have $p(\theta)$ apriori desnsity function about unknown parameter $\theta$, and we consider an optimization problem as follows: \begin{equation} \max_{P \in \mathcal{M}} \int_{\...
0
votes
1answer
87 views

1D functional equation: solve for function with given expected value w.r.t normal density

Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation $$ \begin{split} \mathbb ...
0
votes
0answers
89 views

Convexity of the Frobenius norm of the product of matrices

I have a question similar to Convexity of the Frobenius norm of the product of two matrices. I am not able to comment on that question as I don't have enough reputation, and that is why I have asked a ...
-1
votes
1answer
74 views

subgradient calculus [closed]

I have to calculate a subgradient of the following function $$ p(x) = \max\{e(Cy-Cx)\ :\ Cy \geq Cx, y \in X\}, $$ where $C$ is a $p\times n$ matrix and $X$ is a convex polyhedral set. It is a non-...
5
votes
1answer
207 views

On faces of convex sets of positive semidefinite matrices

A face $F$ of a convex set $C$ is a set such that, if $x \in F$ is a convex combination of other elements in $C$, then they must also be in $F$. I will denote by $F(C,x)$ the smallest face of $C$ ...
1
vote
0answers
39 views

In convex optimization we know that the optimum solution is on which hyper plane

We have a standard linear program, I mean a set of inequalities $c_i^Tx\leq b_i$ where $i\in \{1,\ldots ,k\}$ and we want to find $max\{c^Ty| y\in \{\cap \{x|c_i^Tx\leq b_i\}\}$. I put some condition ...
-1
votes
1answer
95 views

Does a half plane contain intersection of some other half planes? [closed]

I'm doing research in Optimization and I have found this obstacle in the way. If we have set of half planes like $c_ix\leq b_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better ...
1
vote
0answers
32 views

How is the minimax oracle used to find the oracle complexity of projected subgradient?

I am going through a set of blog posts on the complexity of projected gradient method. https://blogs.princeton.edu/imabandit/2013/03/15/orf523-oracle-complexity-large-scale-optimization/ defines the ...
3
votes
0answers
99 views

Weak to weak$^*$ continuity of the duality mapping

Let $X$ be a uniformly convex and uniformly smooth Banach space. We consider the duality mapping $J_p^X$ defined as the sub-gradient $\partial (\frac1p\|\cdot\|^p)$. Is there a characterisation of the ...
1
vote
1answer
67 views

Norm of solution of quadratic program

In a quadratic program (QP), do linear equality constraints always reduce the norm of the minimizer? Specifically, let $P \succ 0$, $A \in \mathsf{M}_{m\times n}$ and $q\in\mathbb{R}^n$. Define $$x^* ...
4
votes
2answers
241 views

Is this function always bounded below?

Is there any global constant $C$ such that $$C<\frac{\sum_{i=1}^{n}x_{i}\log x_{i}-x_{i}+(1-x_{i})\log(1-x_{i})}{\sum_{i=1}^{n}x_{i}}+\log(\sum_{i=1}^{n}x_{i})-\log(\sum_{i=1}^{n}x_{i}^{2})$$ for ...
6
votes
2answers
138 views

Inequality on permutation polytope

Let $a = (a_1, \cdots, a_n), b = (b_1, \cdots, b_n), c = (c_1, \cdots, c_n) \in \mathbb{R}^n$ with $a_1 \geq \cdots \geq a_n, b_1 \geq \cdots \geq b_n, 0 < c_1 \leq \cdots \leq c_n$. In addition, ...
3
votes
2answers
170 views

Mixtures of log-convex functions are log-convex: a reference

A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...
2
votes
1answer
194 views

example of convexity

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $(S_1,S_2)\in \mathcal{B}(F)^2$. We define $$W(S_1,S_2)=\{(\langle S_1 y\; ,\;y\rangle,\langle S_2 ...
4
votes
0answers
121 views

Sharp constant for inequality with convex functions

This is a follow up to this question, where the optimal constant was left open. Let $P \subset \mathbb{R}^n$ be bounded, convex, and open. Let \begin{equation} \mathcal{H} := \{f : P \rightarrow \...
2
votes
0answers
32 views

Do internal stable sets contain big manifolds?

Given two strictly concave functions $u_{i}$ with continuous derivatives in $\mathbb{R}^{k}$. We define their upper levels at a point $x$ of these functions as the set of points y such that $u_i(y)>...
0
votes
0answers
13 views

characterize a family of convex function parametrized by maximal monotone sets

To start, a set $G\in \mathbb R^2$ is called maximal monotone if it is not a strict subset of a monotone set. For each maximal monotone set $G$, we can define a function $$u_G(x_1,x_2)=\inf_{y\in G} \{...
3
votes
0answers
44 views

Behavior of vector field that is the arg min of a convex function

Say $f(x,y)$ is a real valued function that is strongly convex and twice differentiable in both $x$ and $y$. Here $x$ and $y$ are real vectors of different dimensions. We define $y^\star(x)= \arg \...
0
votes
0answers
60 views

Interpretation and upper bound for a convex body projection measure

Let $\mathcal T^t(d)$ be a symmetric convex body of diameter $d$ at origin in $\Bbb R^t$ and let $\delta>0$ be a real number. Pick $2m$ vectors $$v_1,\dots,v_m\in \mathcal T^t(2)$$ $$u_1,\dots,u_m\...
12
votes
1answer
380 views

Average measure of intersection of a convex region with its translate

Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region. My question is about $$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$ How ...
6
votes
1answer
271 views

Fenchel-Rockafellar Duality in Villani's Book

Villiani writes (some notation changed) in Topics in Optimal Mass Transportation: Theorem 1.9. Let $E$ be a normed VS, $E^*$ it topological dual. $\Theta$ and $\Psi$ are two convex functions on $E$ ...
1
vote
1answer
47 views

Calculate $k:=\sup\left\{\left\|\theta\right\|_{*} \: |\: \ell^{*}(\theta)<\infty\right\}$ for a special $\ell$ function

We consider the function $\ell:\mathbb{R}^{m}\rightarrow \mathbb{R}$ given by $$\ell(\xi):=-\max\left\{-\left\langle x,\xi\right\rangle+10 \tau, -51\left\langle x,\xi\right\rangle -40\tau \right\}$$ ...
1
vote
2answers
118 views

Let $X\subseteq\mathbb{R}^n$ and let $F$ be a face of $\mathop{\rm conv} X$. Then $F=\mathop{\rm conv}(X\cap F)$

This theorem is obviously true if the set $X$ is finite (so that $\mathop{\rm conv} X$ is a convex polytope). I believe it is true for any set $X\subseteq\mathbb{R}^n$ but I cannot prove it. Can ...
0
votes
0answers
22 views

Failure of Fenchel duality when algebraic interior condition is met

Consider a proper, lower semicontinuous function $f$ on a Hausdorff locally convex vector space and a convex set $C$ such that $\mathrm{dom}f\cap\mathrm{core}(C)\neq\emptyset$, where core denotes ...
2
votes
0answers
87 views

Covering a space by cones

Let $X\subset\mathbb{R}^n$ be some connected and bounded $n$-dimensional manifold, e.g. a space homeomorphic to an open/closed ball with possibly some parts of it being removed. I am interested in ...
1
vote
1answer
55 views

Properties of Relative Entropies

I'm looking at a paper by Arnold et al. (CPDE, 2001), in which they make use of convex functions in the context of relative entropies. There, they assume that on $(0,\infty)$, their entropy function ...
2
votes
0answers
35 views

Mean width of intersection of two elipsoid

My question is regarding mean widths. For a set $\mathcal{T}$ define the mean width \begin{align*} \omega(T)=\mathbb{E}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I})}\bigg[\underset{\mathbf{u}\in\mathcal{...
3
votes
1answer
119 views

closure of a separating set of pure states

Let $A$ be a unital C*-algebra, and let $\mathcal R$ be a separating family of irreducible representations of $A$. Each vector state of a representation in $\mathcal R$ is a pure state, and the span ...
1
vote
0answers
18 views

Find conditions over $U$ such that optimization over a convex cone generated by $U$ is equal to optimization over $U$ [duplicate]

Reading several pappers to prepare my thesis I found the following problem: We considerer the following optimization problem $$ \left\{\begin{array}{cl} \max\limits_{x\in\mathcal{C}} & f(x) \\[...
8
votes
1answer
512 views

property of convex functions

I am able to give a proof to the following inequality for convex functions. Most likely this is well known, but I am unable to find a reference. I would appreciate if someone more knowledgeable in the ...
1
vote
0answers
76 views

Does the convex-hull of a set contain zero (II)?

Let $(\lambda_1, \cdots , \lambda_d) \vdash k$ be a partition of $k$ of length $d$. Suppose we have $k$ non-zero vectors $\alpha^{(j)} = (\alpha^{(j)}_1, \cdots , \alpha^{(j)}_d) \in \mathbb{Z}^d$, ...
1
vote
1answer
178 views

The tightest upper bound on $-\left\langle B(y-x),\nabla f(A x)\right\rangle$

Let $f$ be a $C^2$ convex function with Hessian $H$ such that $lI\mathcal\le H(x)\le LI,\ \forall x\in\mathbb{R}^n$. Then using Taylor's theorem it follows that $$f(x)-f(y)+\frac{l}{2}\|y-x\|_2^2\le-\...
7
votes
0answers
294 views

Geometry of level sets of a convex function

EDIT: Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $f\colon \Omega\to\mathbb{R}$ be a function such that for some $\lambda$ the function $f(x)+\lambda |x|^2$ is convex. Assume that the ...
2
votes
0answers
64 views

Optimization over a convex cone generated by a set is equal to optimization over the set

Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows: We considerer the following optimization problem $$ \left\{\begin{array}{cl} \...
0
votes
0answers
236 views

Proof of quasi-concavity

I want to prove that a function $f(x)$ is quasi-concave. Unfortunately $f(x)$ is complicated and requires a lot of parameters. Therefore I can't use the following rule: $$ f(λu+(1−λ)v≥min[f(u),f(v)]$$...
1
vote
2answers
118 views

Separating convex sets in Vector spaces

This question just popped on my mind. Let $A, B$ two disjoint, nonempty convex sets in the vector space $X$, can they be separated via a nonzero linear function in $X' = \{ f : X \to R ~ | \quad \...
1
vote
0answers
90 views

Strong smoothness of Lp norm

I asked this question in math.stackexchange but got no answers (link: https://math.stackexchange.com/questions/2323520/strong-smoothness-of-lp-norm). So I decided to ask this question here. Hope I ...