Questions tagged [convex-analysis]

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Generalized Jensen's inequality for positively homogeneous functions

The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
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2 votes
1 answer
94 views

Convex series and closed convex hulls in normed spaces

Let $(X, \lVert \cdot \rVert)$ be a normed space over $\mathbb{R}$ and $A = \{ a_1,a_2 \ldots \} \subseteq X$ be a closed bounded set. Let $\overline{\mathrm{co}}(A)$ denote the closed convex hull of ...
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3 votes
0 answers
86 views

Does smoothing a non-log-concave distribution make it more log-concave?

Suppose that $p$ is a density on $\mathbb{R}^d$ that is $C^2$ and nonzero everywhere, and such that the Hessian of its negative logarithm is lower bounded: $$-\nabla^2 \ln p\succeq L$$ for some matrix ...
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2 votes
0 answers
31 views

Convex optimization over compact sets defined as Aumann set-valued integrals

Let $(X,P)$ be a probability measure space. Let $K$ be a convex compact subset of $\mathbb R^d$ and let $F:X \to 2^{K}$ be a set-valued map. Assume that $F$ is: closed (i.e $F(x)$ is closed for ...
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2 votes
1 answer
93 views

Multivariate inequality of floor function

Define $$f(x,a) := (2x-a)\lfloor\frac{x}{a}\rfloor-a\lfloor\frac{x}{a}\rfloor^2.$$ It seems that $$f(x,a)+f(x,b)\geq 2f(x,c),\forall a,b \in [1,x],a+b=2c.$$ I have written a program that has checked ...
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1 vote
0 answers
26 views

Support functions for subset and superset

I have an ellipse $\mathcal{E} = \{x^TAx = 1\}$, and I have a connected subset of an ellipse $U\subset \mathcal{E}$ For a given $\theta$ let $x_U^*(\theta) = \arg \sup\{\langle x,\theta\rangle, x\in ...
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1 vote
0 answers
16 views

Convex body with prescribed normals (i.e. the Gauss map of the boundary)

Let $\mathbf{S}$ be the unit Euclidean sphere in $\mathbf{R}^n$. I write $u \bullet v$ for the scalar product of two vectors and $A \sim B$ for the set-theoretic difference of sets. Assume $g : \...
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2 votes
0 answers
87 views

Convergence in Hausdorff distance of intersection of closed linear subspaces with a given closed convex set

I've run into the following problem when doing some work with non-commutative metric spaces, which seems like something people may have thought about before but I can't find anything on this problem ...
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  • 21
1 vote
0 answers
54 views

Maximizing a parametric integral over the unit sphere

I am trying to compute the nonnegative quantity $$ \underset{y\in\mathbb{S}^{d-1}}{\sup}\int_{0}^{t}(\Vert A(\tau)y\Vert_{1}- \Vert A(\tau)y\Vert_{q})d\tau, \quad 1 < q < \infty $$ where $\...
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125 views

Is this function concave? If it is, can we show it in a theoretical way?

Suppose we have a function: \begin{equation} \begin{aligned} f(x_1,x_2,\cdots,x_n)&=\sum\limits_{i=0}^n\frac{(x_i e^{-x_i}-x_{i+1}e^{-x_{i+1}})^2}{e^{-x_i}-e^{-x_{i+1}}}\\ &=\frac{(x_0 e^{-x_0}...
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1 vote
2 answers
142 views

Can we substitute this KKT condition into this optimization problem to reformulate the optimization problem?

Suppose I have the following optimization problem $$ \min\limits_{\mathbf{x},\mathbf{y}} f(\mathbf{x},\mathbf{y}) \tag{1} $$ It is already known that the target function $f$ is continuous and ...
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  • 81
1 vote
1 answer
125 views

Is a convex, lower semicontinuous function that is bounded from below, actually continuous?

While thinking about convex functions, I managed to put together the following proof which I find a bit too good to be true. $X$ is a topological vector space that is also a Baire space. Lemma: Let $f ...
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  • 509
1 vote
1 answer
119 views

Convex/concave points of a differentiable function

I am wondering about the following question: A strictly convex (concave) differentiable function $f:\mathcal{R}\to\mathcal{R}$ has the geometrical property that its graph lies completely above (below) ...
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  • 481
0 votes
1 answer
94 views

Estimation via projecting onto a convex body

Assume that $\theta$ is in a convex body $K \in \mathbb{R}^n$ and we observe $y = \theta + z$, where $z$ is a noise term (following, say, the normal distribution). Consider an estimator of $\theta$ by ...
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8 votes
1 answer
545 views

Is the square root of the Kullback-Leibler divergence a convex map?

$\newcommand{\KL}{\operatorname{KL}}$Let $X$ be a Polish metric space and $P(X)$ the space of probability measures on $X$. Given $\mu, \nu\in P(X)$, recall that $$\KL(\mu\parallel\nu) = \begin{cases}\...
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  • 231
3 votes
1 answer
124 views

On the convexity of certain set of random vectors

Let ${\cal X}$ be the set of pairs of random variables $(X,Y)$ with finite expectations. For constant $c\in[0,1]$, define set $$ \{(X,Y)\in{\cal X}:\exists a\geq 0, \, b\geq 0 \text{ such that } E[\...
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4 votes
1 answer
232 views

Which set of functions admits the existence of the minimizer?

Let $a,b \in \mathbb R$ and consider the functional $J$ on $X$: $$J[u] = \int_0^1 \left( (u'(x))^2 -a)^2 + b \ln (1+ u^2(x))\right) dx$$ Providing reasons specify if the $\inf J$ over $X$ is attained ...
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0 votes
0 answers
51 views

Projection onto a cone followed by a Schur-convex function

Let $Proj_C(x)$ denote the projection of a point $x$ onto a cone $C$. Let $f$ be a Schur-convex function. I'm considering $f(Proj_C(x))$ as a function of $x$. Are there any conditions on the cone $C$ ...
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1 vote
0 answers
24 views

Sufficient condition for an $n$-tuple to be a convex conjugate

We say $(f_1,f_2,\dotsc,f_N)$ is a convex conjugate if for any $i=1,2,\dotsc,N$ and any $x_i\in\Bbb R^d$, we have: $$f_i(x_i)=\sup\left\{\sum_{k=1}^{N}\sum_{j=k+1}^N x_k x_j - \sum_{j=1,j\neq i}^N f_j(...
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2 votes
1 answer
97 views

Smoothness of Minkowski functional is equivalent to smoothness of boundary

Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$, $$ f_C(x):=\inf\Big\{t>0:\frac1{t}\cdot x\in C\Big\}, $$ is $C^1$ ...
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1 vote
1 answer
174 views

Proof of extended version of non-random "almost supermartingale"

In this question, a non-random version of "almost supermartingale" theorem is proved. Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder ...
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1 vote
1 answer
67 views

Sufficient conditions for the boundary unit-normal vector field of a closed convex set to be Lipschitz continuous

This is followup to the following question: On the Lipschitz continuity of the unit-normal vector field of a polytope Let $C$ be a (nonempty) closed convex subset of $\mathbb R^n$. Note that to every ...
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1 vote
1 answer
162 views

Can we invoke "almost supermartingale" Theorem for deterministic sequences?

Perhaps stupid question. Question: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems? Attempt ...
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2 votes
1 answer
99 views

On the Lipschitz continuity of $x \mapsto \arg\min_{c \in C}d(x,c)$ w.r.t Hausdorff distance

Let $C$ be a (nonempty) compact subset of euclidean $\mathbb R^n$, and consider the set-valued map $p_C:\mathbb R^n \to 2^C$ defined by $$ p_C(x) = \{c \in C \mid \|x-c\| = \mbox{dist}(x,C)\}, $$ ...
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  • 5,580
1 vote
1 answer
68 views

On the Lipschitz continuity of the unit-normal vector field of a polytope

Let $\mu$ be a probability measure on $\mathbb R^n$ and let $P$ be a compact polytope in $\mathbb R^n$. For any $x \in \mathbb R^n \setminus P$, let $p(x) \in P$ be (unique!) point in $P$ which is ...
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1 vote
1 answer
96 views

Conditions for Lipschitzness of boundary normal vector, almost everywhere

Let $C$ be a nonempty closed subset of $\mathbb R^n$. It is known that any such set satisfies the following condition (Unique CPP a.e). For almost every $x \in \mathbb R^n$, there exists a unique ...
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2 votes
0 answers
60 views

A truncated Frobenius norm of a matrix is convex or not?

Given a positive integer $k$ and a matrix $X\in \mathbb{R}^{m\times n}$. A truncated frobenius norm of a matrix $X$ is defined by $$\Vert X \Vert_{k,F} = \sqrt{\sum_{i=k+1}^{m} \sigma_i^2(X)},$$ where ...
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3 votes
1 answer
180 views

Well-behaved trajectories

Call trajectory any continuous function $f: \mathbb{R}_{\geq 0} \to \mathbb{R}^n$ (here, $\mathbb{R}_{\geq 0}$ is interpreted as time). A polyhedral partition of $\mathbb{R}^n$ is a finite set of ...
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  • 141
4 votes
0 answers
151 views

Legendre-Fenchel transform

Suppose $F:\mathbb R^n\to \mathbb R$ is a convex continuous function. Moreover, for any $x\in \mathbb R^n$, $$ \limsup_{\lambda\to\infty} \frac {|F(\lambda x)|}{\lambda}<\infty. $$ I would like to ...
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3 votes
0 answers
65 views

Projection onto level set of convex functional

Fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $F:L^2_{\mathbb{P}}(\mathcal{F})\rightarrow (-\infty,\infty]$ be bounded-blow, convex, lower semi-continuous, and not identically ...
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4 votes
1 answer
179 views

On some convergence theorems by Felix E. Browder (1967)

I have been reading Felix E. Browder's Convergence Theorems for Sequence of Nonlinear Operators in Banach Space and I was hoping I could find answers to a couple of questions I have about the paper. ...
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1 vote
1 answer
49 views

Measure of intersection of convex set with hyperplane is concave function

Let $\Omega \subset \mathbb{R}^n$ be convex. We write points of $\mathbb{R}^n$ as $(x_1, x_2, \dots, x_n)$. Set $p(x) = m(\Omega \cap \{x_1 = x\})$, where $m$ is the $n-1$ dimensional Lebesgue measure ...
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4 votes
2 answers
263 views

Convexity of $(X, y) \mapsto y^T X^{-1} y$ [closed]

Let $y \in \mathbb{R}^n$, $X \in \mathcal{S}^n_{++}(\mathbb{R})$. Why would function $ f : (X, y) \mapsto y^T X^{-1} y$ be convex? I tried with $(X, x) + t.(Y, y)$ with no result. Also, I thought ...
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2 votes
1 answer
157 views

Bounded variation of the partial derivatives of a convex function

Let $f:\mathbb R^2\to\mathbb R$ be a convex function. For simplicity, assume that $f\in C^1$. A general theorem which can be found in the book of Evans and Gariepy says that the gradient $\nabla f$ is ...
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1 vote
1 answer
44 views

Elementary inequality about integrals of exponentials of concave functions (possibly connected to log concave distributions)

I think the following inequality might be true and was hoping somebody might spot it or know a proof: Suppose $f:\mathbb R\to \mathbb R$ is convex and suitably nice so that $$\int_{\mathbb R} e^{-f(x)}...
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0 votes
0 answers
47 views

An inequality regarding operator concave function

Crossposted from math.SE Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $...
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  • 141
2 votes
0 answers
91 views

About Hilbert-Haar theory

The Hilbert-Haar theory says that functionals $\mathcal{F}(u,B)=\int_{B} F(\nabla u)\,dx$, where $F$ is a convex function and $B$ is a bounded domain in $\mathbb{R}^N$, take a minimum in the space of ...
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0 votes
1 answer
82 views

Log-concavity of the modified Bessel function of a second kind

I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified ...
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0 votes
0 answers
40 views

Subgradients of $|Du|(\Omega)$ and Hodge decomposition in dual spaces of measures

Let $\Omega \subset \mathbb R^d$ be open and bounded. I have a function $u\in BV(\Omega)$ and a vector field $\mu \in L^\infty(\Omega)^d$ such that $-div\ \mu\in L^2(\Omega)$, and $-div\ \mu$ is in ...
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  • 256
4 votes
0 answers
59 views

Length of curves on convex hypersurfaces

Let $\gamma\colon[a,b] \to \mathbb{R}^n$ be a smooth curve. Let $f_i\to f$ be a sequence of convex functions on $\mathbb{R}^n$ converging uniformly on compact subsets to $f$. Let $\hat\gamma(t):=(\...
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  • 19k
1 vote
1 answer
154 views

How to prove the convexity of a simple function involving a ratio of two polygamma functions?

Let \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*} and $$ \psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}. $$ In the literature, ...
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  • 390
1 vote
1 answer
102 views

Can the subdifferential become unbounded at interior points?

Consider $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ a lower-semicontinuous, proper, closed and convex. My question is, can the subdifferential of $\partial f$ be unbounded in the interior of $\text{...
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  • 141
2 votes
0 answers
52 views

Biconjugate of a quasiconvex lower semi-continuous function

Let $f:\mathbb{R}^d \to [0,\infty]$ be a quasiconvex lower semi-continuous function whose effective domain $C:=\{x \in \mathbb{R}^d:f(x) < \infty\}$ is nonempty and bounded (and convex since $f$ is ...
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  • 139
7 votes
1 answer
190 views

A question on regularity of the Legendre transform

Let $f(x)$ be a strictly convex real-valued $C^{\infty}$ function on an open neighborhood of the origin in $\mathbb R^n$ with $f (0, \ldots , 0)= \partial_j f(0, \ldots , 0)=0$ for all $j$. If the ...
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  • 545
0 votes
1 answer
121 views

When is a convex function continuous on its domain?

Consider a lower-semicontinuous convex function $f\colon \mathbb{R}^n \to \mathbb{R}$ with domain $C = \{x \in \mathbb{R}^d: f(x) < \infty\}$. I am interested in understanding under what conditions ...
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2 votes
1 answer
129 views

Analytic expression for the Moreau envelope of $x \mapsto \|Ax\|$

Given an $m \times n$ matrix $A$ and a vector $c \in \mathbb R^n$, define $\eta(A,c) \ge 0$, $$ \eta(A,c) := \sup_{u \in \mathbb R^n} u^\top c - \frac{1}{2}\|u\|^2 - \|Au\|. $$ Note that $\eta(A,c) = \...
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  • 5,580
2 votes
0 answers
130 views

Convexity of volume in terms of a deformation

In the context of convex optimization and mechanics, I am interested in the convexity of the potential energy $U$ of a pressure acting over some volume $V$ enclosed by a surface. Here pressure can be ...
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  • 51
0 votes
0 answers
52 views

When is the optimal value of a problem convex in the constraint parameter?

Let $f:(0,1] \to [0,\infty)$ be a $C^2$ strictly decreasing function, $f(1)=0$. Define $$ F(s):=\min_{xy=s,x,y\in(0,1]} f(x)+ f(y). $$ Question: Can we characterize the functions $f$ which give rise ...
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  • 6,286
6 votes
1 answer
95 views

Lower convex envelope of polynomial functions

Let $P\in{\mathbb R}[X]$ be a polynomial and $[a,b]$ be a bounded interval. Of course, the graph of $P$ is an algebraic set. I am interested in the lower convex envelope $\bar P$ of $P|_{[a,b]}$. It ...
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  • 47.1k
4 votes
1 answer
137 views

Is Axiom of Choice for convex sets of distributions on naturals necessary?

Take any family $(S_i)_{i∈I}$ such that each $S_i$ is a convex set of functions $f : ℕ→[0,1]$ where $\sum_{k∈ℕ} f(k) = 1$. By "convex" we mean that for any $f,g∈S_i$ and any $a,b∈[0,1]$ such ...
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