Questions tagged [convex-analysis]

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How large are the smallest-area projections of a high-dimensional convex body?

Let $B$ be a convex body in $\mathbb{R}^d$, equipped with its standard Euclidean form, and assume that $$\intop_B x \, dx = 0$$ $$\frac{1}{|B|_d} \intop_B x_i x_j \, dx = \delta_{ij},$$ a ...
Alexander Shamov's user avatar
11 votes
1 answer
502 views

A function with unexpectedly simple Legendre transformation

Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and \begin{equation} I(x)= \begin{cases} \frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\ \...
minhtoan's user avatar
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9 votes
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A self-isometry of the sphere of a strictly convex Banach space that does not move basic vectors

Problem. Let $n\in\mathbb N$, $X$ be a strictly convex $n$-dimensional real Banach space, $S_X=\{x\in X:\|x\|=1\}$ be the unit sphere of $X$, and $e_1,\dots,e_n\in S_X$ be linearly independent points. ...
Taras Banakh's user avatar
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9 votes
0 answers
910 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
8 votes
0 answers
192 views

Concavity of product and ratio of sums

Apologies if this question is not appropriate for MathOverflow. I have asked at Math.StackExchange without success. Consider the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ defined as $$ f(x)=\...
user_lambda's user avatar
7 votes
0 answers
400 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:...
Piotr Hajlasz's user avatar
7 votes
0 answers
869 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 ...
asv's user avatar
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7 votes
0 answers
230 views

orthogonal projector onto the set of convex functions

Let $\Omega\subset \mathbb R^d$ be an open, convex domain, and consider the Hilbert space $L^2(\Omega)$. Each sum of convex functions is convex, hence the subset $Conv(\Omega)$ of all convex functions ...
Delio Mugnolo's user avatar
6 votes
0 answers
105 views

"Moduli space" of isotropic convex bodies?

A lot of questions in convex geometry revolve around the geometry of isotropic convex bodies in $\mathbb{R^n}$. To my knowledge there is no, or very little study of a space such as : $$C_n = \{...
Gericault's user avatar
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5 votes
0 answers
133 views

Conv A = Dual B

I have two cones $A$ and $B$ in a Euclidean space. I want to show that $\mathrm{Conv}\,A=\mathrm{Dual}\,B$; that is, $a\in\mathrm{Conv}\,A$ if and only if $\langle a,b\rangle\ge 0$ for any $b\in B$. ...
Anton Petrunin's user avatar
5 votes
0 answers
182 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. ...
Jairo Bochi's user avatar
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5 votes
0 answers
210 views

Concavity of mixed volumes and mixed discriminants

For $n\times n$ symmetric matrices $A_1, \ldots, A_n$, the mixed discriminant $D(A_1, \ldots, A_n)$ can be defined as $1/n!$ times the coefficient of $t_1\ldots t_n$ in the homogeneous polynomial $\...
Sasho Nikolov's user avatar
5 votes
0 answers
867 views

Decreasing sequence of closed convex sets in a Banach space

Let $(C_n)$ be a decreasing sequence of closed convex subsets in a Banach space $(E,\Vert \cdot \Vert)$. The question I have is about the content of $C=\bigcap_{n=0}^\infty C_n$. If the $C_n$ are ...
mathcounterexamples.net's user avatar
4 votes
0 answers
130 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{...
Motaka's user avatar
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4 votes
0 answers
355 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
  • 499
4 votes
1 answer
220 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 ...
Holden Lee's user avatar
4 votes
0 answers
222 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 ...
user12345678's user avatar
4 votes
0 answers
63 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):=(\...
asv's user avatar
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4 votes
0 answers
124 views

A convex function is "usually" subdifferentiable

Let $X$ be a locally convex topological vector space, and let $f:X\to\mathbb R\cup\{\infty\}$ be a proper, convex, lower semicontinuous function, whose effective domain $D:=f^{-1}(\mathbb R)$ is ...
e.lipnowski's user avatar
4 votes
0 answers
63 views

A standard name of a strongly extremal point of a convex set

I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the ...
Taras Banakh's user avatar
  • 40.8k
4 votes
0 answers
83 views

Elementary functions (growing faster than exponential ones) with elementary Legendre–Fenchel transforms

Let $F$ be the set of all convex functions $f\colon[0,\infty)\to[0,\infty)$ with $f(0)=0=f'_+(0)$ and $f_+(\infty-)=\infty$, where $f'_+$ is the right derivative of $f$. For any function $f\in F$, its ...
Iosif Pinelis's user avatar
4 votes
0 answers
252 views

Can this function be minimized?

Let $X$ be a locally convex TVS, and let $A$ and $B$ be convex and compact subsets of $X$ with $A \subset B$. Let $f: A \times B \to [0,\infty]$ have the following properties: (1) For all $b \in B$, $...
aduh's user avatar
  • 839
4 votes
0 answers
183 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 ...
leo monsaingeon's user avatar
4 votes
0 answers
332 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 ...
Christian's user avatar
  • 779
4 votes
0 answers
200 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 \...
Steve's user avatar
  • 1,085
4 votes
0 answers
202 views

Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$. ...
Iosif Pinelis's user avatar
3 votes
0 answers
57 views

Convex combination of cyclically monotone sets

I want to show the following statement, but I am not sure how. Proposition(?): Let $C \in \mathbb{R}^d$ be a compact convex set, and let $u, v : C \to \mathbb{R}$ be smooth convex functions. Suppose $$...
Paruru's user avatar
  • 51
3 votes
0 answers
98 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), \...
πr8's user avatar
  • 688
3 votes
0 answers
156 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 ...
P. Quinton's user avatar
3 votes
0 answers
85 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 ...
ABIM's user avatar
  • 5,019
3 votes
0 answers
46 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 ...
M. Reza. K's user avatar
3 votes
0 answers
101 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 ...
John's user avatar
  • 483
3 votes
0 answers
768 views

Do the gradient of convex (Fenchel) conjugates preserve the "distance" between two uniformly convex functions?

Update: I have discontinued pursuing this question. However, by observing that the conjugate and its derivative are nothing more than optimum and optimizer, my question should be answered by carefully ...
Nick's user avatar
  • 31
3 votes
0 answers
94 views

Legendre transform on signed measure space

Let $X$ be an open set in $\mathbb{R}^n$ and $M(X)$ be the space of finite signed measures defined on $X$. $L(p)$ is a lower-semicontinuous convex functional defined on $M(X)$. My question is: (1) ...
Elliott's user avatar
  • 325
3 votes
0 answers
223 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 \...
good bandit's user avatar
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
3 votes
0 answers
142 views

Probability of hitting two vectors

Call an element of $ \{-m,\dots,0,\dots,m\}^{2^n}$ a vector. Assume $m = O(2^{2^n})$. Let $u_1,u_2$ be vectors. Let $\{v_1,\dots,v_{2^n}\}$ and $\{w_1,\dots,w_{2^n}\}$ be linearly independent ...
Turbo's user avatar
  • 13.7k
3 votes
0 answers
179 views

Characterization of minimizer of convex functional

I need to check whether the following characterization of the minimizer of a convex functional is valid. Let $X$ be a reflexive Banach space (think $W^{1,p}(\Omega)$ with $\Omega \subset \mathbb R^n$ ...
D G's user avatar
  • 201
3 votes
0 answers
229 views

Area defined with $\pm$ closedness

Denote $B_n\subset\Bbb R^n$ to be unit ball at origin. Denote $S\subset B_n$ to region of type $\mathsf I$ if it satisfies $$s\in S\iff\forall t\in S, s+t\in S\mbox{ or }s-t\in S$$ I am convinced $\...
Turbo's user avatar
  • 13.7k
3 votes
0 answers
381 views

On increasing the penalty term in convex optimization with regularization

Given the two strictly convex (unique solution) optimization problems as: $$Problem\:1:\min_{X} f(X)+\|X\|_{F}^2 \hspace{2cm}Problem \:2:\min_{X}f(X)+n\|X\|_2^2$$ where $X\in\mathbf{S}_{++}^{n}$ (...
borntotry83's user avatar
3 votes
0 answers
74 views

Minimax-like theorems involving union and intersection of regions in $\mathbb R^d$

From Minimax theorems, we roughly know that if $f(x,y)$ is convex on $X$ and concave $Y$ for both compact $X,Y$, then: $$ \max_{x\in X}\min_{y\in Y} f(x,y)=\min_{y\in Y}\max_{x\in X} f(x,y). $$ We can ...
Arash's user avatar
  • 722
3 votes
0 answers
338 views

Convex functions with non-singular hessian measure are continuously differentiable?

It is known that every convex function $f: \Omega\to \mathbb{R}$, $\Omega$ convex subset of $\mathbb{R}^n$, has a weak derivative of bounded variation $Df\in BV_{loc}(\mathbb{R}^n)$ (e.g. Evans and ...
Ettore Minguzzi's user avatar
3 votes
0 answers
499 views

Directional derivates and unique subgradients

I have a question about the fine structure of convex functions. Convex functions behave very regular in the interior of their domain of definition (e.g. they are locally Lipschitz continuous there) ...
Dirk's user avatar
  • 12.3k
3 votes
0 answers
253 views

Ways to establish equality of measures on locally compact spaces

Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality $$ ...
Appliqué's user avatar
  • 1,269
3 votes
1 answer
59 views

Looking for an Alternative Characterization of the Upper Orthant of a Convex Hull

I have a question on convex analysis, which I require in one step of my research work. Before stating it, let me give a small background. Let $Y, X_1,\ldots,X_n$ be $n+1$ points in $\mathbb{R}^d$. ...
Usermath's user avatar
2 votes
0 answers
46 views

Instances of c-concavity outside of optimal transport?

Let $X$ and $Y$ be metric spaces, and let $c:X\times Y\rightarrow \mathbb{R}$ be a nonnegative function which we refer to as a cost. For any $\phi:X\rightarrow \mathbb{R}$ and $\psi:Y\rightarrow \...
Brendan Mallery's user avatar
2 votes
0 answers
55 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 $...
dohmatob's user avatar
  • 6,716
2 votes
0 answers
120 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 \...
theouscidda6's user avatar
2 votes
0 answers
73 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 ...
dohmatob's user avatar
  • 6,716
2 votes
0 answers
46 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 : \...
Sławek Kolasiński's user avatar