Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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If we inscribed all the 6 regular convex four-dimensional polytopes in a sphere, which one would have the highest volume?

When a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%). But what about for the 6 regular convex ...
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221 views

Relationship between convexity and superadditivity?

This question is a little vague, so let me give some motivation. I was trying to prove the generalized Holder's inequality for probability measures, $$\mathbb{E}(X_1 \dots X_n) \leq \prod_{i=1}^n ...
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Let $S$ be a closed convex set & $x$ be an extreme pt of $S$ then $S-\{x\}$ is

Let $S$ be a closed convex set and $x$ be an extreme point of $S$, then $S-\{x\}$ is Convex Not Convex May or may not be convex I am thinking that the convexity doesn't fail even if we remove the ...
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What are the formulas for the number of vertices, edges, faces, cells, 4-faces, …, $n$-faces, of convex regular polytopes in $n \geq 5$ dimensions?

I know that in dimension $n \geq 5$ there are only 3 kind of convex regular polytopes in each dimension: the $n$-simplex, the $n$-cube and the $n$-orthoplex. I would like to know if there are ...
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257 views

How do I prove that the composition of an affine function preserves convexity?

What would be the formal proof that $ f(Ax + b) $ given $ f(x) $ is a convex function ? I got to the point where I expanded $$ f(\lambda(Ax+b) + (1- \lambda)(Ay+b)) = f(A(\lambda x + (1 - \lambda)y) + ...
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Adjacency of convex hull facets

Let $C$ be a $d$-dimensional convex polytope and $p$ is a point outside of it. $C=\{f_c\}$ defined by set of facets $f_c=\{p_c,A_c\}$ where $p_c$ is a tuple of vertices and $A_c$ is a set of ...
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28 views

Example of a uniformly convex domain in $\Bbb R^n$

I am trying to understand the differences between a convex domain, and a uniformly convex domain. Intuitively, to my knowledge, a convex domain is one where any line between any two points in the ...
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Infinite dimensional convex cones

Let $C$ be a convex cone in a topological real vector space $V$. Assume that we have a linear functional $\varphi: V \to \mathbb{R}$ such that $\varphi(x) \geq 0$ for all $x \in C$. Further assume ...
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24 views

Nominalization for being “not convex” and “not coercive”

Having a function f which is neither convex nor coercive. What is the correct nominalization for these properties? I suppose something like ...
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38 views

Lower semicontinuity of a Bochner integral of a convex function

I'm looking for the following result: Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $f$. The map $$u \mapsto \int_0^T \int_{\Omega} f(u(t))$$ is lower semicontinuous for $u \in ...
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Theorem about convex sets

I want to prove a theorem (the link is after the whole text here) and in order to do that I need to prove three preliminary statements. I tried to prove them all but I'm already stuck in the first, ...
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21 views

Detecting faces of polytopes

I am working in convex geometry for the summer with little experience beforehand. It's a lot of fun but it does mean I don't know some of the basic things. I'm interested in the orbits of finite ...
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1answer
55 views

uniqueness of Hahn-Banach extension for convex dual spaces

Let $X'$ be strict convex, i.e. for all $x_1',x_2'\in X'$ with $\|x_1'\|_{X'}=\|x_2'\|_{X'}=1$ the implication $$\left\|\frac{x_1'+x_2'}{2}\right\|=1\Rightarrow x_1'=x_2'$$ holds. In this case the ...
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about an interesting affirmation involving convex sets

Consider the following definition Definition: Let $\Omega \subset R^n$ a bounded convex set. A point $x \in \partial \Omega$ is called an extremal point if $x$ cannot be written as linear ...
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Volume of a polytope with relaxed constraints

Consider a polytope in $n$ dimensions defined by a set of linear constraints: $$P = \{x \in \mathbb{R}^n : Ax \leq b\}$$ where A is some $m \times n$ constraint matrix, and $b = (b_1,\ldots,b_m)$ is ...
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Calculating Legendre Transform

Let $\Omega,\Omega^*$ be bounded domain in $\mathbb{R}^n$, $B_r(y_0)$ is a small ball of radius $r$ center at $y_0\in\Omega^*$, in $\Omega^*$ define a function $\psi(y)=-\sqrt{(r^2-|y-y_0|^2)}$ on ...
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Difference quotient inequality with convex functions

A convex function on an interval $ I $ is said to be convex if for every $ 0 < t < 1 $ and $ x,y \in I $ we have that $ f(tx + (1-t)y) \le tf(x) + (1-t)f(y)$. Prove a function is convex if and ...
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33 views

Cone of convex solutions

I have been reading a paper, on monge-ampere type equations, and the existence of a unique convex solution has been proven to exist in $C^{3,\alpha}(\Omega)\cap C^{2,\alpha}(\overline{\Omega})$, for ...
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extension of Legendre transform of a function to a larger domain

Let $\Omega,\Omega^*$ be bounded domain in $\mathbb{R}^n$ and $u_0$ be a uniformly convex function define on $\Omega$. Suppose the gradient of $u_0$ maps $\Omega$ into a subdomain of $\Omega^*$, i.e. ...
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56 views

Regularization vs. Inequality Constraint

For what values of a regularization parameter $\alpha$, there is an equivalent inequality constraint in convex optimization? In particular, in the convex optimization problems below $$ \text{ Problem ...
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Proving convexity

I ask you please some help with this problem: Let A $\subseteq$ R$^n$ be a convex set and $C(A)$ = {$\lambda$x, $\lambda \in \mathbb{R}$, $\lambda \geq 0$, x $\in$ A}. Prove that $C(A)$ is convex. ...
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Is it possible to convexify this cone constraint?

General question An SOCP constraint is given by: $$ \| A_i \mathbf{x} + b_i\| \leq \mathbf{c}_i^T \mathbf{x} + d_i.$$ I have the following constraint: $$ \| A_i \mathbf{x} \| \geq d_i.$$ Is it ...
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A a.e. strongly convex function

Suppose that $f=f(x)$ is strongly convex a.e. for $x\in\mathbb{R}$, i.e. there exists $\epsilon>0$ such that $f''(x)\geq\epsilon>0$ a.e. for $x\in\mathbb{R}$. Then there exists ...
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Dual of dual cone of nonconvex closed cone

let $K$ be a nonconvex closed cone, then $K^{**}=conv(K)$ should this hold? I am not quite sure about it. Thanks.
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34 views

Linear constraint in convex optimization

Is it true that the solution to a linearly constrained convex minimization problem can only be placed on the boundary of the constraint set, for any nonlinear convex objective, e.g. $$ \min_x f(x)$$ ...
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53 views

Rate of convergence of an exponential function

If I have a function $$f = \exp(\sqrt{n} \cdot \frac{\sqrt{\log{n}}}{\sqrt{n}-\sqrt{\log n}}),$$ I can notice, that $$\lim_{n \to \infty} f = \infty,$$ but also I can notice that it goes very slowly ...
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29 views

Supremum of convex lipschitz functions.

Let $f_i:K\to R, i\in I$ be a family of convex, equi-Lipschitz functions on some compact subset $K$ of $\Bbb R^n$. Is it true that $\sup f_i$ is also Lipschitz continuous(assuming that the sup ...
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1answer
62 views

Prove that function is convex

How can it be proved that the function $f(x) = \ln \bigl(\sum\limits_{i=1}^{n} e^{x_{i}}\bigr) $ is convex?
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property of Convex body on the plane

Let $K$ be a convex body on the plane with smooth curve. Observe the triangle $\triangle ABC$ that contains $K$ with minimal perimiter and let $X,Y,Z$ the points on $BC,AC,AB$ that belong to $K$. I ...
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Show that $\mathcal{C}(A)$ is the smallest convex of $E$ containing $A$.

Let $E$ be a $\mathbb{R}$-vector space, and $A$ a nonempty subset of $E$. Show that $$\mathcal{C}(A) = \biggl\{\sum \limits_{k=1}^n \lambda_kx_k \biggm| n \in \mathbb{N}^*, (x_1,\dots,x_n) \in ...
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compact and convex set

I recently have worked on compact convex sets in the context of time series and my question is related to that. If we have a set $$ C=\{\beta_1 X_1 +\beta_2 X_2 +\phi H_1 , ...
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Projection onto Polyeder

I know how to projects onto a linear subspace of $\mathbb R^3$, but how to project a point $x$ onto an polyhedron given as the intersection of three halfspaces $$ \langle y_1, x \rangle \ge c_1 ...
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Series and concavity

If $u(x)$ is strictly concave, can I say: $$ \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^{n+1}\cdot u(n) < \infty. $$ I am having trouble finding counterexamples. Thanks.
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Convexity defined by Karamata inequality

Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions?
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Convex optimization: affine equality constraints into inequality constraints

I have the following problem: \begin{equation} \begin{array}{cll} \displaystyle \min_{ \mathbf{x} } & & \displaystyle f(\mathbf{x}) \\ \mathrm{s.t.} & & \mathbf{x} \in \mathcal{C} \\ ...
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Separate two convex sets with disjoint interior (in $\mathbb{R}^n$)

In $\mathbb{R}^n$, I know that if $A$ is a convex set and $b$ in the boundary of $A$. Then we can separate $A$ and $b$, which means there exists $f \in \mathbb{R}^n$ such that $f\cdot x \ge f \cdot b$ ...
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How can the function's composite be convex function?

$h$ is a continuous function which is convex and strictly decreasing; $t$ is a continuous function which is strictly increasing; $\operatorname{ran}(t)$ is included in $\operatorname{dom}(h)$. Under ...
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regarding the concept of dual cone

When studying the covex analysis, I am not clear about the concept of dual cone. In the following graph, $\mathcal{K}*$ was the dual cone. I marked two points, the ...
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Properties concave functions

Is is true that if $f(x)$ is a concave function of $x$ with domain $C$, then $f'(a) \leq \frac{f(a)}{a}$ for any $a \in C$, where $f'(a)$ denotes the derivative of $f(x)$ with respect to $x$ evaluated ...
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if convex or nonconvex function

There is an iteration recurrence relations between the argument. In fact, it is part of my optimization model . The equation F is convex or not convex? thank u
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119 views

Convex subsets, Normed spaces, Separating hyperplane

I'm trying to understand the proof of a theorem taken from a textbook. Theorem: If $S_1,S_2$ are disjoint non-empty convex subsets of a real vector space $X$ (which may be infinite dimensional) then ...
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Polar set and adjoint operator

I'm trying to understand the following statement: A bounded operator between Banach spaces $u:X\rightarrow Y$ satisfies: $$\textrm{inf}\{\;||u(B_X \cap S)||:S\subset X,\,\textrm{codim}S=k \; ...
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Convexity in each argument and directional derivative

Let $f(x,y)$ be a continuous function, convex in each argument separately. Does this imply the existence of one-sided directional derivatives in any direction? For example, does there exist (finite or ...
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To prove the existing and uniqueness of a solution

Let function $f$ be differentiable and convex in $R^{n}$. How can it be proved that $\forall \lambda > 0$ solution of system equations $f'(x) = -\lambda x$ exists exclusively ($\exists \hspace{3mm} ...
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Separating convex sets in a tvs $X$.

I got doubt with the proof of this theorem. Let $X$ be a tvs, $A,B \subset X$ with $A$ an open convex set and $B$ convex such that $A \cap B = \emptyset$. Then there exists $f \in X^*$ (where ...
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Union of all sets of optimal solutions to a perturbed linear programming problem

Please let me know if you have some ideas on how to approach this proof? I got stuck part-way through. The following linear program is a function of $\theta$, $ \begin{array}{ll} \min & c^\top x ...
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Convexity of a region on probability simplex

Exercise 2.15 g of Boyd et al Convex Optimization book : On the probability simplex in $\mathbb{R}^n$ where each point $p = (p_1,p_2,p_3,\ldots,p_n)$ corresponds to a distribution for random variable ...
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Proving function defined by algorithm is convex

I'm working on my bachelor thesis and I'm trying to prove a conjecture, but I seem to miss the hint that helps me. I have an algorithm that defines a function $f:\mathbb{R}_{\geq ...
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Characterization of Convex Polygons

John Lee's Axiomatic Geometry has an interesting characterization of convex and non-convex vertices for polygons. Let $P$ be a polygon. Consider a ray emanating from a vertex of $P$ which does not ...
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Concavity of a function

While I am reading a book I couldn't follow the following step. " By concavity of the function $x \sqrt{\log\frac{1}{x}}$ for $x \in (0,1)$ we have that " $O(x \sqrt{\log\frac{k}{x}})$ = ...