# Questions tagged [convex-analysis]

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311
questions

**4**

votes

**1**answer

112 views

### Is there a non-convex function with non-decreasing average rate of change?

$\newcommand{\R}{\mathbb R}$
Let $f$ be a function from $\R$ to $\R$. It is said that $f$ is midpoint-convex if for any real $x$ and $y$ we have $f(x-y)+f(x+y)\ge2f(x)$ or, equivalently, $f(x+y)-f(x)\...

**2**

votes

**0**answers

41 views

### Separability of Minkowski Sum of well-behaved sets

Let $A$ and $C$ be non-empty simply connected and connected subsets of $\mathbb{R}^k$ and suppose that $C$ is convex. Then is the Minkowski sum $A+C$ separable?

**1**

vote

**1**answer

63 views

### $\arg\max$ in the dual norm of the nuclear norm

Given a matrix $X \in \mathbb{R}^{m \times n},$ then the spectral norm is defined by
$$\left \| X\right\| := \max\limits_{i \in \{1, \dots, \min\{m,n\}\} }\sigma_i (X)$$
whereas the nuclear norm is ...

**2**

votes

**0**answers

36 views

### A complete metric space with some convex-type property

Let $(X,d)$ be a complete metric space with this property:
for each $x∈X$, $r>0$ and $y∈X$ with $d(x,y)<r$, there exists $z∈X$ such that $d(x,y)+d(y,z)=d(x,z)=r$.
I want to know if the family ...

**0**

votes

**0**answers

16 views

### Derivation of dual for infinite linear program

I'm reading the section on Linear Programming in Barbu and Precupanu's Convexity and Optimization in Banach Spaces, and had a couple of questions concerning their derivation of the dual problem for ...

**2**

votes

**0**answers

41 views

### improved regularization for $\lambda$-convex gradient flows

It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...

**0**

votes

**1**answer

50 views

### Concavity of a function along a path

Suppose that $f(x,y)$ is a continuously differentiable function
and $g(x,y) =xy-f(x,y)$. I know that $g$ is concave
if and only if $(-f_{xx})(-f_{yy}) -(1-f_{xy}) ^{2}>0$ and $f_{xx}>0$.
Now ...

**5**

votes

**1**answer

85 views

### When is the log-permanent concave?

Let $\operatorname{PSD}_n$ be the cone of $n\times n$ semidefinite positive matrices. For any $X\in \operatorname{PSD}_n$, define $$f(X)=\log(\det(X)).$$ Then $f$ is a concave function on $\...

**1**

vote

**0**answers

81 views

### Given a finite set of points, does there exist a linear function pass through a point and strictly below the other points for all the points?

I guess my question is a follow up question of this one: usul, Existence of a strictly convex function interpolating given gradients and values, version: 2019-04-13.
In usul's question, the answer ...

**0**

votes

**0**answers

27 views

### Extended-value subgradients

(I am not an expert in convex analysis, as may become clear.)
Let $R$ be the extended reals, $R = \mathbb{R} \cup \{\pm \infty\}$.
Standard texts define the subgradient of a convex function $f: \...

**3**

votes

**0**answers

70 views

### Implicit function theorem for subdifferentiable convex functions

I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...

**0**

votes

**1**answer

126 views

### Finding the conjugate of a function

I know that the Fenchel conjugate of a function is
$$f^*(x^*) = \sup_x\{\langle x, x^*\rangle - f(x)\}.$$
However, how do I find the Fenchel conjugate of the function
$$f(x) = \frac{1}{p}\sum\limits_{...

**2**

votes

**1**answer

69 views

### Convergence of semi convex functions

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...

**2**

votes

**1**answer

242 views

### When does $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$ hold for a real function $f(x,y)$?

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality:
$$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$
For ...

**5**

votes

**1**answer

176 views

### Generalization of minimal selection theorem

Consider a metric space $X$ and a set-valued map $F : X \to \mathbb{R}^{n}$. We define the minimal selection
\begin{equation*}
m(F(x)) := \arg\min \big\{ \lvert u \rvert : u \in F(x) \big\},
\end{...

**1**

vote

**0**answers

38 views

### Subgradient chain rule

Suppose $$F:\mathbb{R}^n \to \mathbb{R},\; F(x)=\mathrm{max}_\mathrm{eig}(C-\mbox{diag}(x)).$$
I am trying to find a subgradient of $F$ at $x_0$. A subgradient of $\mathrm{max}_\mathrm{eig}$ is given ...

**2**

votes

**0**answers

45 views

### Variational forms of non-convex functions

I am trying to understand what kind of variational forms exist for non-convex functions. Alternatively, are there conjugate forms which attain strong duality? For a non-convex function $f$, I am ...

**11**

votes

**1**answer

330 views

### Aleksandrov's proof of the second order differentiability of convex functions

Aleksandrov [A], proved a remarkable property of convex functions.
Theorem. If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and ...

**9**

votes

**0**answers

226 views

### Second order differentiability of convex functions

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is ...

**2**

votes

**1**answer

53 views

### Inequalities for upper semi-continuous affine functions on compact sets by using extreme points

Suppose $f_1\colon K\to [0,\infty)$ and $f_2\colon K\to[0,\infty)$ are two upper semi-continuous affine functions,
$$
f_i(\lambda x+(1-\lambda)y)=\lambda f_i(x)+(1-\lambda)f_i(y)\ \mbox{ for all }\ x,...

**1**

vote

**1**answer

63 views

### Extreme points of an intersection of convex set with countably many linear spaces

Let $V$ be some `nice' vector space and let $T: V\to \mathbb{R}$ be a linear functional over $V$.
Define
\begin{align}
M= K \cap \bigcup_{i \in \mathbb{N} } \{ v \in V: T(v)=c_i \}
\end{align}
...

**1**

vote

**1**answer

133 views

### Log-concavity of function

Consider the function
$$f_{n}(x)=e^{-x^2}x^n.$$
My goal is to show that
$$ G(y):=\frac{(f_2*f_0)(y)}{(f_0*f_0)(y)}- \left(\frac{(f_1*f_0)(y) }{(f_0*f_0)(y)}\right)^2$$
is log-concave.
Let us ...

**1**

vote

**2**answers

98 views

### Monotonicity of maximum of convex combination of two scaled concave functions

Let $f:R\rightarrow R$ be a concave function with a unique and finite maximum. Let $$g(x, \beta) = \beta f(\alpha \cdot x) + (1-\beta) f((1-\alpha) \cdot x), $$ where $\alpha \in [0,1/2]$ and $\beta \...

**3**

votes

**0**answers

54 views

### subgradient in a predual under weak* continuity

Let $X$ be a Banach space. Suppose $f:X^*\to\mathbb R\cup\{\infty\}$ is convex, has closed and bounded (and so weak*-compact) effective domain, and is weak*-continuous on its effective domain. In ...

**0**

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94 views

### How are the $L^2$ and $\sup$ norms related on the space of strongly convex functions?

Given a convex compact set $X \subset \mathbb R^d$ with interior containing the orign let $V$ be the space of all smooth functions $f: X \to \mathbb R$ with the properties:
The function is strongly ...

**0**

votes

**0**answers

30 views

### Characterization of global subdifferentiability

Let $X$ be a locally convex space, $D \subseteq X$ a nonempty compact convex set, and $f: D\to\mathbb R$ a continuous convex function.
Question: Is there any known, interesting, alternative ...

**1**

vote

**0**answers

193 views

### Continuity of the Legendre transform of a Lipschitz function

Let $p > 1$, $f$ defined on $L^p(\mathbb{R}^n,\mathbb{R})$, locally Lipschitz and concave, with $f(x) = 0$ when $x \geq 0$ a.e. We define, with $q$ dual to $p$, for any $y\in L^q$ , $g(y) := \sup_{...

**2**

votes

**0**answers

52 views

### Prove the equicontinuity of a maximizing sequence

Let $X$ be a compact subset of $\mathbb{R}$ and $c(x_1,x_2,x_3,x_4)$ be a fixed bounded continuous functions on $X^4$. Assume $\mu,\nu$ are probability measures on $X^2$, and $\mu\otimes\nu$ is the ...

**7**

votes

**1**answer

278 views

### Does midpoint-convex imply rationally convex?

Say that a function $f$ defined on $\mathbb{Q}^n$ is midpoint convex if $f((x+y)/2) \le (f(x) + f(y))/2$. Say that it is rationally convex if, for $\lambda \in \mathbb{Q} \cap [0,1]$ and $\bar \lambda ...

**1**

vote

**1**answer

166 views

### Most general form of Jensen's inequality

What is the most general form of Jensen's inequality?
Wikipedia gives for example this more general form, which holds in every topological vector space.
Are there even more general forms, for ...

**0**

votes

**0**answers

53 views

### Proximity operator composition

Suppose that $f,g$ are convex and lower-semi continuous functions from $\mathbb{R}^d$ to $\mathbb{R}$ such that $f\circ g$ is lsc and convex. I know that the proximity operators $\operatorname{Prox}...

**5**

votes

**0**answers

153 views

### Generic shadows of convex bodies

If $K \subset \mathbb{R}^m$ is a convex body (i.e., a compact convex set) and $T \mathbin\colon \mathbb{R}^m \to \mathbb{R}^n$ is a linear map then $TK \subset \mathbb{R}^n$ is a convex body as well. ...

**3**

votes

**1**answer

102 views

### What is a non-trivial example of an unbounded subdifferential?

Let $f: X \to [ -\infty, \infty]$ be some function,
Can someone provide a non-trivial example where the subdifferential evaluated at a point $x$,
$$\partial f(x)$$ is "unbounded"? (trivial examples ...

**0**

votes

**1**answer

217 views

### Expectations, double integrals and Jensen's inequality

$\def\anonfunc#1{#1(\cdot)}$Consider two random variables distributed $v\sim \anonfunc G$ and
$c \sim \anonfunc F$ with pdfs $\anonfunc g$ and $\anonfunc f$. Let the supports of $c$ and
$v$ be $[x,y]$....

**3**

votes

**0**answers

66 views

### Sufficient condition for convex conjugate to be second-order differentiable

Let $f:\mathbb{R}^n\to (-\infty,\infty]$ be a convex lower-semicontinuous function, we then define its conjugate by
$$
f^*(y)=\sup_{x\in \mathbb{R}^n}\{x^Ty-f(x)\}.
$$
Then there exist well-known ...

**18**

votes

**2**answers

1k views

### A strange variant of the Gaussian log-Sobolev inequality

Let $\phi : \mathbb{R}^d \to \mathbb{R}$ be a convex function, and assume that it grows at most linearly at infinity for simplicity. Denote by $\gamma$ the standard Gaussian measure on $\mathbb{R}^d$, ...

**3**

votes

**1**answer

68 views

### On the area-perimeter ratio of a convex limited set

(Previously asked on MSE)
Let $C\subset \mathbb{R}^2$ be a convex limited set. We define the average radius of $C$ as
$$a_C=\frac{\int_{v\in C}d(v,C)dxdy}{A(C)}$$
Where $d(v,C)$ is the distance ...

**0**

votes

**2**answers

192 views

### Goldowsky-Tonelli theorem for upper semi continuous function

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a convex and continuous function. We know that $\partial^{e} f(. )$(either right or left derivative) is non-decreasing and upper semi-continuous function. ...

**1**

vote

**0**answers

47 views

### How does one translate from convex hull to a set of facets (inequalities)? [duplicate]

Suppose I have defined a convex set as the convex hull of a set of points. (I know that all these points are "extremal points" of the convex set.) I know want to translate this description of the ...

**1**

vote

**1**answer

62 views

### Separation in $l^1$ (Kreps-YanTheorem)

I have a question about the hypotheses of the Kreps-Yan Separation Theorem. I use the notation $l^p_+$ for the subspace of vectors all of whose coordinates are non-negative and define $l^p_- = -l^p_+$....

**2**

votes

**0**answers

98 views

### Maximization of an integral functional over a closed convex set

I want to maximize $$F(w):=\sum_{1\le i,\:j\le2}\int\lambda^{\otimes2}({\rm d}(x,y))\left(w_i(x)f_j(x,y)\wedge w_j(y)f_i(y,x)\right)g_{ij}(x,y)$$ over the closed convex set $$S:=\left\{w\in{\mathcal L^...

**1**

vote

**0**answers

61 views

### Gradient formula for Clarke's generalized gradient on a general Banach space

In Theorem 10.27 of the book Functional Analysis, Calculus of Variations and Optimal Control, there is the following gradient formula:
($\operatorname{co}$ deotes the convex hull).
Is there an ...

**1**

vote

**1**answer

121 views

### How can we calculate the generalized gradient of $L^2\ni x\mapsto a\min(x(s),by(t))$?

Let $(T,\mathcal T,\tau)$ be a measure space, $a,b\ge0$, $s,t\in T$ and $$f(x):=a\min(x(s),bx(t))\;\;\;\text{for }x\in L^2(\tau).$$
How can we calculate the generalized gradient $\partial_Cf(x)$ of ...

**0**

votes

**3**answers

98 views

### Intersection of Subharmonic and Harmonic Functions (also: Extension of harmonic functions)

Let $\Omega \subset \mathbb{R}^n$ be open, convex and bounded with smooth boundary. Define
$$\mathcal{A}(\Omega) = \left\{ C \subset \Omega \ \big\vert \begin{array}{l} \text{for any subharmonic ...

**3**

votes

**1**answer

256 views

### When do convexity and lower semicontinuity imply continuity?

Let $X$ be a nonempty compact convex subset of a locally convex space. Say $X$ has the convex function property if every convex, lower semicontinuous $f:X\to\mathbb R$ is also continuous.
Question: ...

**4**

votes

**1**answer

223 views

### Given convex l.s.c. function $f$, find decreasing convex function $\phi$ such that $f(x) \equiv \sup_y x\phi(y)-\phi(-y)$

Let $f: \mathbb R \rightarrow (-\infty,+\infty]$ be a lower-semicontinuous convex function.
Question
Under what futher conditions does there exists a convex decreasing function $\phi: \mathbb R \...

**1**

vote

**0**answers

65 views

### Extreme points of set of measures with given barycenter

Let $X$ be a convex compact metrizable subspace of a locally convex Hausdorff topological vector space, $x_0\in X$, and $P$ be the space of all Borel probability measures on $X$ with barycenter $x_0$.
...

**2**

votes

**1**answer

216 views

### Does the Legendre-Fenchel transform/convex conjugate of strongly convex functions have any desirable properties?

It is well known in convex analysis that when a closed, proper, function $f$ is Legendre-type, that is, essentially strictly convex and essentially smooth, the Legendre transform yields a dual ...

**0**

votes

**0**answers

36 views

### Example of a sequence of logarithmically convex functions on $\mathbb{R}$ and for all $n\in\mathbb{N}$ in the spirit of one evoked in an article

To ask this question I was inspired in some words, if I understand well, from the authors of a preprint on arXiv in section 4.1, that I believe that is [1], to ask next question.
We consider the ...

**0**

votes

**0**answers

24 views

### Edit: Minimizing sequence of a multi-valued function using proximal point algorithm

The following is the multi-valued function I want to minimize: $f_1:\mathbb{R}^2\rightarrow \mathbb{R}$ defined by:
$$f_1(x_1,x_2)=100x_1^2,\tag{1}$$
for all $x=(x_1,x_2)\in \mathbb{R}^2.$ The ...