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18 votes
1 answer
3k views

How bad can the second derivative of a convex function be?

One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property: $$\label{p}\tag{P} f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
Tomás's user avatar
  • 409
9 votes
0 answers
978 views

Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
Mary's user avatar
  • 91
7 votes
1 answer
856 views

Compactness of set of indicator functions

Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set $$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$ Is this set compact in $L^\infty(0,1)$ with respect ...
Saj_Eda's user avatar
  • 395
7 votes
0 answers
249 views

Proving this function is convex

Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
Tom Solberg's user avatar
  • 4,049
4 votes
2 answers
561 views

Reference request: concave/convex envelope

I'm seeking the references concerning on the regularity analysis of concave envelopes, i.e. given some measurable function $f:\mathbb R^d\to\mathbb R$ that is bounded from above ($d=1$ or $d\ge 1$), ...
CodeGolf's user avatar
  • 1,835
4 votes
1 answer
317 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 \...
dohmatob's user avatar
  • 6,853
4 votes
0 answers
481 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}...
Nik Bren's user avatar
  • 519
3 votes
1 answer
171 views

Characterization of a set in $\mathbb{R}^d$

Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set. \begin{equation}\label{main12} C= \{x\in \mathbb{R}^d ~|~ ...
Math123's user avatar
  • 57
3 votes
0 answers
63 views

Is the collection of Schur convex functions sequentially compact?

We know in ROCKAFELLAR's convex analysis chap 10 that the collection of uniformly bounded convex functions on compact set is sequentially compact. I wonder if it is still true for the collection of ...
Sung-En Chiu's user avatar
2 votes
1 answer
197 views

Prékopa-Leindler style inequality?

Does anyone know a simple proof of the following Prékopa-Leindler style inequality: If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$,...
Anthony's user avatar
  • 125
2 votes
1 answer
255 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,...
Iosif Pinelis's user avatar
2 votes
1 answer
141 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 ...
jason's user avatar
  • 553
1 vote
1 answer
156 views

On the additive property of the subdifferential of lower semicontinuous functions

Let $f:\mathbb R\to\mathbb R$ be a lower semicontinuous function, we define the Fréchet subdifferential of $f$ at $x\in\mathbb R$ by $$\partial^F f(x):=\left\{L\in\mathbb R: \liminf_{v\to0}\frac{f(x+v)...
Fergns Qian's user avatar
1 vote
2 answers
359 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. ...
Adam's user avatar
  • 1,043
1 vote
1 answer
114 views

Reference request: regularity of functionals on the space of probability measures

Let $\mathcal M=\mathcal M(\mathbb R^d)$ be the space of finite measures on $\mathbb R^d$, and $\mathcal P=\mathcal P(\mathbb R^d)\subset\mathcal M$ be the space of probability measures. Let $F:\...
CodeGolf's user avatar
  • 1,835
1 vote
0 answers
79 views

Conditions on triangle inequality for integral kernel

Consider $\int_RK(x,y)f(y)dy$, where $K(x,y) \in M_+(R^2)$. Let $L(t,s)$ be an iterated rearrangement of $K$. Let also $$ A(t,v)=\int_0^{1/v}L(1/t,s)ds, $$ which is decreasing with $v$ and ...
user124297's user avatar
1 vote
0 answers
303 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_{...
Alfred's user avatar
  • 31
1 vote
0 answers
78 views

Potential for a Monotone Operator

[Cross-posted from math.stackexchange] I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The ...
Pete Caradonna's user avatar
1 vote
0 answers
389 views

The perturbation of a convex function can also be convex?

$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...
High GPA's user avatar
  • 263
1 vote
0 answers
123 views

Generalization of concave envelope

Let $g:\mathbb R_+\to\mathbb R$ be a measurable function (which could be supposed to be bounded and Lipschitz if required). Let $\mathcal P$ be the collection of probability measures $\mu$ on $\mathbb ...
CodeGolf's user avatar
  • 1,835
1 vote
0 answers
145 views

convergence of supergradient

Let $\{g_n\}$ be a sequence of concave functions defined on $\mathbb{R}$ and set $$\lambda_n(x)=\lim_{\Delta x\to 0+}\frac{g_n(x+\Delta x)-g_n(x)}{\Delta x}$$ Assume there exists a concave function ...
CodeGolf's user avatar
  • 1,835
1 vote
0 answers
217 views

convergence of concave envelope

Let $\{f_n\}$ be a sequence of uniformly upper bounded functions defined on $\mathbb{R}$ s.t. for every $x\in\mathbb{R}$ $$f_n(x)\to f(x),~ n\to\infty$$ Define $g_n$ and $g$ as the concave envelope ...
CodeGolf's user avatar
  • 1,835
0 votes
1 answer
279 views

When does strict inclusion holds for the domain of subdifferential?

Recall that, given an extended real-valued function $f: \mathbb{R}^n \to (-\infty, \infty]$ Its effective domain is, $$\text{dom}(f) = \{x \in \mathbb{R}^n : f(x) < +\infty\}$$ The subdifferential ...
Shamisen Expert's user avatar
0 votes
0 answers
80 views

Verifying the Cauchy behavior of a sequence

Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
PPB's user avatar
  • 85
0 votes
1 answer
603 views

A concave function as supremum of upper semi continuous is upper semi continuous

We define a affine(concave), upper semi continuous function and bounded function $f:X \to \mathbb{R}$, where $X \subset \mathbb{R}^{k}$ is compact and convex set. Assume that $T$ is an affine and ...
Adam's user avatar
  • 1,043