# Questions tagged [convex-analysis]

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454
questions

2
votes

1
answer

59
views

### Smooth approximation of nonnegative, nondecreasing, concave functions

Let $f\colon [0, \infty)\to\mathbb{R}$ be nonnegative, nondecreasing, and concave. Prove the following claim or give a counter example: There is a sequence of functions $f_n\colon [0, \infty)\to\...

10
votes

1
answer

318
views

### Convex functions in convex sets

Suppose $\Omega \subset \mathbb{R}^n$ is some bounded, convex set. For which domains $\Omega$ is it true that for every convex function $f:\Omega \rightarrow \mathbb{R}$ the average of the function in ...

3
votes

2
answers

82
views

### Subdifferential of a convex function admits a continuous selection

Let $F$ be a continuous convex function on $\mathbb{R}^n$.
If the subdifferential $\partial F(x)$ of $F(x)$ admits a continuous selection, for every $x \in \mathbb{R}^n$, does it mean that $F$ is ...

0
votes

0
answers

100
views

### Characterization of a "complex" hull?

This is a complex continuation of my previous question. There Iosif Pinelis showed that the so obtained closure from taking the intersection of the preimages of the linear functionals indeed coincides ...

1
vote

1
answer

55
views

### Viscosity characterization of convex functions

Let $\Omega\subseteq\mathbb{R}^n$ open and convex. It is elementary that if $u\in C^2(\Omega)$ then
$$u \text{ is convex}\iff D^2u\geq0 \ \text{ in } \Omega$$
I was looking for a similar ...

6
votes

2
answers

512
views

### Conditions for including cones

Consider $N$ $n$-dimensional vectors, where the angle between any two vectors is acute and their starting point is at the origin. Can we rotate these vectors together so that the coordinate components ...

2
votes

1
answer

99
views

### Tangent cone of a closed convex cone

Let $K \subset \mathbb{R}^n$ be a closed convex set. Given a point $u \in K$, the tangent cone of $K$ at $u$ is defined as (or characterized by)
$$
T_K(u) := \mathrm{cl}(\left\{ t (v - u) \mid v \in K,...

4
votes

2
answers

117
views

### An upper bound of gradient norm for convex functions near minimizer

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a convex function. Denote $X^*$ as the set of minimizers of $f$ and assume $X^*$ is unbounded. Is it possible that $\|g_x\|$ is unbounded when $d(x,X^*)$ ...

1
vote

1
answer

62
views

### Legendre transformation of vector valued function

Good afternoon. Is there any generalisation of Legendre--Fenchel transformation to the vector-valued functions $f: \mathbb{R}^n \to \mathbb{R}^n$?

5
votes

1
answer

171
views

### Large ideally convex sets

Let $E$ be a Banach space. A set $C \subseteq E$ is called ideally convex if for every bounded sequence $(x_n)$ in $C$ and for every sequence $(\lambda_n)$ in $[0,1]$ that sums up to $1$ the vector $\...

0
votes

0
answers

42
views

### On convexity of special fractals in the plane

Let $X \subset \mathbb{R}^2$ be a subset of the plane with Hausdorff dimension $1<dim_H(X)<2$.
For a subset $Y \subset \mathbb{R}^n$ we define $Y$ to be convex if for every $y_1,y_2 \in Y$ the ...

0
votes

0
answers

80
views

### Decomposing convex functions as (simple, useful, convex) + (convex)

Let $f$ be a nice convex function, let $X = \{ x_i : i \in [N] \}$ be a collection of points in the domain of $f$, and define the 'bundle approximation'
$$
f(x; X) = \max \{ f(x_i) + \langle \nabla f (...

7
votes

1
answer

478
views

### Asking for an English version of Aleksandrov's famous 1939 paper in Convex Geometry

I have difficulty even in finding a Russian version of the next paper:
"Aleksandrov, A. D., Almost everywhere existence of the second differential of a convex
function and some properties of ...

0
votes

0
answers

35
views

### Solving constrained convex minimization problems by reduction to a scalar equation

Let $f,g:\mathbb R^d \to \mathbb R$ be functions such that
$g$ is convex and there exists $x_0 \in \mathbb R^n$ such that $f(x_0) \le 0$.
$f$ is strictly convex.
Consider the problem
$$
\tag{*}
\...

0
votes

0
answers

73
views

### The study of directional derivatives for functions that are minimums of convex functions

Has research been conducted on the topic of directional derivatives of functions that are minimums of convex functions? It would be greatly appreciated if you could provide the appropriate reference.
...

2
votes

0
answers

51
views

### Whether $d_x(t) := \|P_t(x)-x\|_H$ is increasing in $t$ where $P_t:H \to H$ is the proximal operator of a convex function

Let $H$ be a Hilbert space (e.g Euclidean $\mathbb R^n$), and fix a proper convex function $f:H \to (-\infty,+\infty]$. Given any $t \ge 0$, let $P_t:H \to H$ be the proximal operator of $f$ at level $...

0
votes

0
answers

55
views

### Support function of the intersection of a hyper-ellipsoid and a Euclidean ball

Le $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_d$ be positive numbers. For any $x \in \mathbb R^d$ and $r \ge 0$, define $\gamma(x,r) := \sup_{z \in E(r)}x^\top z$, where
$$
E(r) := E \cap B_2^d(r)...

0
votes

0
answers

78
views

### Bound for the laplacian of a strictly convex function from above by the gradient of it

Let $V \in C^2(\mathbb{R}^d; \mathbb{R})$ a (strictly) convex function with $ \int_{\mathbb{R}^d} \mathrm{e}^{-V(x)} \, dx = 1.$
I am trying to show that
$$ \int_{\mathbb{R}^d} |\nabla_x V(x) |^2\...

3
votes

0
answers

80
views

### Convex minorants to convex functions, given partial Taylor expansion and smoothness estimate

Let $V$ denote a strictly convex function (in arbitrary dimension) whose Hessian is $L$-Lipschitz. Given only this knowledge, and the values of $\left\{ V \left( x \right), \nabla V \left( x \right), \...

1
vote

1
answer

74
views

### Strict convexity of the cost function is enough to ensure the existence and uniqueness of the optimal transport map

Let $X=Y = \mathbb R^d$ and $c:X \times Y \to [0, \infty)$ be Borel measurable. Let $\mu, \nu$ be Borel probability measures on $X,Y$ respectively. Let $\mathcal T$ be the set of all Borel measurable ...

1
vote

1
answer

129
views

### Is a Lipschitz continuous gradient equivalent to this condition?

I know if a function $f: \mathbb{R}^n \to \mathbb{R}$ is $L$-smooth, i.e. its gradient $\nabla f$ is $L$-Lipschitz continuous, then it satisfies the following inequality for any $x, x_0 \in \mathbb{R}^...

5
votes

1
answer

382
views

### Is the Legendre transform as an operator Lipschitz?

Let $C_{lsc}(\mathbb{R}^n)$ be the space of lower semicontinuous convex functions $\mathbb{R}^n \to \mathbb{R}$. The Legendre-Fenchel (LF) transform of $f \in C_{lsc}(\mathbb{R}^n)$ is:
$$ f^*(y) := \...

4
votes

0
answers

125
views

### A Lipschitzian's condition for the measure of nonconvexity

I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by:
$$\alpha(A)=\sup_{b\in\overline{\operatorname{...

7
votes

0
answers

323
views

### Are there any characterizations of $C^2$ convex functions?

There are several characterizations of convex functions with the Lipschitz continuous gradient. If we already know that the function is of class $C^1$, then we have the following equivalent conditions:...

0
votes

1
answer

57
views

### Strict inclusion for recession cone of closure of a convex set

Let $C$ be a nonempty closed convex subset of $\mathbb{R}^n$. The recession cone of $C$ is given by
$$R_C=\left\lbrace d\in\mathbb{R}^n:x+td\in C, \forall t>0, \forall x\in C\right\rbrace.$$
It is ...

1
vote

0
answers

127
views

### The image of zero-measure set under normal mapping is Lebesgue measurable

Let $u$ be a convex function defined on a bounded open set $\Omega$ in $\mathbb{R}^n$. Then $u$ is twice differentiable a.e. Let $E_u$ be the set on which $u$ is not twice differentiable. Then $E_u$ ...

2
votes

1
answer

184
views

### On the infimal convolution of two norms on $\mathbb R^n$

$\newcommand{\R}{\mathbb R}$For natural $n$, $a\in\R^n$, and real $t>0$, let
\begin{equation*}
K:=K_{n,t}(a):=\inf_{x\in\R^n}(\|a-x\|_2+t\|x\|_1),
\end{equation*}
\begin{equation*}
M:=M_{n,...

0
votes

1
answer

60
views

### Is the mapping $F(a):= \arg\min_{x \in \mathbb R^n} \|x-a\|_2 + \|x\|_1$ non-expansive?

Fix $a \in \mathbb R^n$ and let $\|\cdot\|$ be any norm on $\mathbb R$ (e.g $\ell_1$ norm). For any $a \in \mathbb R^n$, it is clear that the function $f_a(x) := \|x-a\|_2 + \|x\|$ is strictly convex ...

0
votes

1
answer

90
views

### Orthogonal projection of a point centrally-symmetric closed convex subset of $\mathbb R^n$ never expands the coordinates of the point

Let $C$ be a closed convex subset of $\mathbb R^n$ which is symmetric about the standard coordinate axes. For example, think of $C$ as the unit-ball for an $\ell_p$-norm, for some $p \in [1,\infty]$. ...

1
vote

0
answers

22
views

### Minimax statistical estimation of proximal transform $\mbox{prox}_g(\theta_0)$, from linear model data $y_i := x_i^\top \theta_0 + \epsilon_i$

tl;dr: My question pertains the subject of minimax estimation theory (mathematical statistics), in the context of linear regression.
Given a vector $\theta_0 \in \mathbb R^d$, consider the linear ...

3
votes

0
answers

128
views

### Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?

I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...

6
votes

2
answers

201
views

### For most directions does the supporting hyperplane meeting a bounded convex set meet it in one point?

Let $C\subseteq \mathbb R^n$ be non-empty, convex and compact. For $v\in S^{n-1}$, let $H_v$ be the supporting hyperplane in the direction of $v$ (i.e., $H_v$ is the boundary of the smallest closed ...

2
votes

1
answer

106
views

### Analytic value of $\alpha := \sup_{(x,y) \in C} ax+by$, where $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$

Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set
$$
\alpha := \sup_{(x,y) \in C} ax + b y.
$$
Question. ...

0
votes

1
answer

61
views

### Analytic formula for minimizer of $f(x) := \sqrt{(x-a)^\top S(x-a)}+ r \|x\|_2$

Let $S$ be a positive-definite $n \times n$ matrix and define $\|z\|_S := \sqrt{x^\top S x}$ for any $x \in \mathbb R^n$. Let $a$ be a fixed vector in $\mathbb R^n$ and $r \ge 0$, and consider the ...

18
votes

3
answers

657
views

### Convergence of convex functions

I can prove the following result.
Theorem 1. Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions
that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$.
Then ...

3
votes

1
answer

144
views

### Probabilistic Taylor theorem for concave functions

This paper proves a probabilistic version of Taylor's theorem
\begin{equation*}
\mathbb{E}g(X) = \sum_{k=0}^{n-1} \frac{g^{(k)}(0)}{k!} \mathbb{E}X^k + \frac{\mathbb{E}X^n}{n!} \mathbb{E} g^{(n)}(X_{(...

2
votes

0
answers

59
views

### Convex ordering of measures that are obtained by different push-forwards of a same measure

Suppose that we have a probability measure $\rho$ which is supported on $\mathbb{R}^d$ and absolutely continuous w.r.t. the Lebesgue measure. Take two vector fields $F, G : \mathbb{R}^d \rightarrow \...

0
votes

0
answers

60
views

### Direct (first-order ?) algorithm to minimize $u(x) := \|x-a\|_C + r\|x\|_p$

Fix $a \in \mathbb R^n$, $r \ge 0$, $p \in \{1,2\}$, and a positive-definite matrix $C$ of order $n$. Define $u:\mathbb R^n \to \mathbb R$ by $u(x) := \|x-a\|_C + r\|x\|_p$, where $\|z\|_C := \sqrt{z^\...

2
votes

1
answer

99
views

### Can we use the solution to two optimisation problems to solve a third, bigger, one?

Background
Say we have an optimization problem $$\min_x f(x) = g(x) + h(x)$$
where $g$ is differentiable and convex, and $h$ are convex but not necessarily differentiable. If $g$ is the mean squared ...

4
votes

2
answers

519
views

### Unit ball of the sum space

Let $V$ be a vector space and $\|\cdot \|_1$ and $\|\cdot\|_2$ two norms on $V$.
Let $\|\cdot\|_+$ be given by
$$ \|v\|_+ := \inf_{v = v_1 + v_2} \|v_1\|_1 + \|v_2\|_2 $$
It is well-known that $\|\...

0
votes

0
answers

45
views

### Intersection of the simplex with a space vector

The sets are defined in $\mathbb{R}_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ will be denoted $\mathring C$. The codimension of a submanifold $M$ of a manifold $N$ will be denoted ...

5
votes

3
answers

465
views

### How to prove this (corollary of) hyperplane separation theorem?

$X$ is a nonempty convex subset of $\mathbb{R}^n$ whose element is $x=\left(x_1,...,x_n\right)$.
The theorem is as follows.
If for each $x\in X$, there is an $i \in \left\{1,...,n\right\}$ such that $...

0
votes

0
answers

62
views

### solution of equivalent problem Kantorovich for case squared distance function

We know that the Kantorovich duality when the cost function is the square Euclidean distance is equivalent to
$$
\inf_{(\tilde\varphi,\tilde\psi)\in \tilde\Phi_c} J(\tilde\varphi,\tilde\psi) = \sup_{\...

5
votes

1
answer

210
views

### Proving the set $\lbrace \frac{(x + y)^2}{\sqrt{y}} \leq x - y + 5, y > 0 \rbrace$ is convex

I have recently picked up a course on Convex Analysis in my spare time, but feel I'm not quite up to speed with the 'tricks' for proving a set is convex.
I have managed to prove this by moving all ...

0
votes

2
answers

325
views

### Smooth approximation for non differentiable function

Let $f(t) = \min(\frac{1}{\lvert t\rvert}, 1)$. I would like to find a smooth approximating function $g$ such that $f(t) \leq g(t)$ for all real $t$. Is there a nice function $g$ out there? Any ...

0
votes

0
answers

48
views

### Minimal value of $f(x) := \|x-x_0\|_S^2 + a\|x-x_0\|_S\|x\| + b\|x\|^2$

Fix $x_0 \in \mathbb R^n$, $a,b \ge 0$ and an $n \times n$ positive-definite matrix $S$. For any $x \in \mathbb R^n$, define
$$
f(x) := \|x-x_0\|_S^2 + a\|x-x_0\|_S\|x\| + b\|x\|^2,
$$
where $\|z\|_S :...

3
votes

2
answers

194
views

### Hausdorff dimension of the non-differentiability set a convex function

Let $X \subset \mathbb R^d$ be open, $f : X \to \mathbb R$ and
$$
E := \{x \in X : f \text{ is not Fréchet differentiable at }x\}.
$$
Then we have the following result which is
Theorem: If $X= \...

-4
votes

1
answer

73
views

### Convex combination of $\frac{1}{x}$ inequality [closed]

Let $0 < x_1 \leq ... \leq x_n$ and $\sum \alpha_i = 1, \alpha_i \geq 0$. Show
$\sum \frac{\alpha_i}{x_i} \leq \frac{x_1 + x_n - \sum \alpha_i x_i}{x_1 x_n} $. Since the left side looks like a ...

3
votes

1
answer

263
views

### Derivative of distance function to a convex set in CAT(0) space

Let $(X,d)$ be a complete CAT(0) space. We denote by $T_x X$ the tangent cone at a point $x\in X$ and by $d_x$ its associated distance. So $(T_x X,d_x)$ is also a complete CAT(0) space. In CAT(0) ...

4
votes

0
answers

233
views

### 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}...