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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Convex polyhedron is union of simplices

Given a convex polyhedron $P$, how can we prove that every point $x \in P$ is in some simplex whose vertices are vertices of $P$? One proof is to inductively build a triangulation of $P$. If $P$ is ...
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Prove that the following two definitions of the convex hull are equivalent.

I was wondering if a topology expert could help me solve this proof, as I have no idea but want to understand these concepts. This is not for homework. Let X be a point set, not necessarily finite, ...
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function defined as an integral involving Bessel functions

i need to analyze a function of the form $$F(x,y) = \int_0^{1} e^{-(1+s)\alpha x}\sinh((1-s)\beta y) I_0(\sqrt{(x^2-y^2)s}) ds $$ Where $I_0$ is the modified Bessel function. $x>y$ always. ...
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Show this function is convex

Show that $f(x)=\frac{(\theta-x)\log_{x}\frac{x-\theta}{1-\theta}+x}{1-x}, ~x\in(\theta,\infty)$ is convex, where $\theta\in(0,1)$. $f(1)=\lim_{x\rightarrow 1}f(x)$. Numerical experiments suggest it ...
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Random convex shapes containing a ball

I'm interested in the properties of randomly generated convex shapes in $n$-dimensional space. Suppose I were to generate $v$ uniformly distributed random points on the $n$-ball of radius $R$. What ...
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238 views

Why does the amoeba shrink to its skeleton when we go to infinity?

Let $f\in\mathbb{C}[X_1^{\pm1},\ldots,X_n^{\pm1}]$ a Laurent polynomial. Let $\mathrm{Log}:(\mathbb{C}\setminus\{0\})^n\to\mathbb{R}^n$ defined by ...
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93 views

Number of faces of convex hull

If you have $n$ points in $d$-dimensional Euclidean space, the number of faces of the convex hull is potentially exponential I understand. How can this be proved?
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400 views

Subdifferential of the sum

Let $C \subset \mathbb R^n$ a nonempty subset. Let us define the indicator function of $C$ $$ I_C(x) = \begin{cases} 0 & x \in C \\ +\infty & x \notin C \end{cases}. $$ Let us consider, in ...
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Analytically solving simple quadratic problem in single variable with boundary constraints

I want to solve the following optimization problem where $x$ is scalar variable. $$ \min_x \dfrac12ax^2 + bx \\ subject\ to:\ l\le x \le u $$ $ a > 0 $ therefore, this is a convex optimization ...
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1answer
41 views

Terminology for a weaked vector space

Let $S$ be a semigroup on which acts $\mathbb{R}_{\geq0}$. Does this structure has a name? For example $S$ can be the set of convex bodies in $\mathbb{R}^n$ with the Minkowsky sum.
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convexity of a function involving inverse matrix

The function has the following form: $f(a_1,a_2,b_1,b_2) = bA^{-1}c$, where \begin{align*} ...
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If $(\nabla f(x)-\nabla f(y))\cdot(x-y)\geq m(x-y)\cdot(x-y)$, why is $f$ convex?

I was reading on wikipedia that a strongly convex function is also strictly convex. I say that a function $f\colon\mathbb{R}^n\to\mathbb{R}$ is convex if $$ f(\lambda x+(1-\lambda)y)\leq\lambda ...
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2answers
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A question about a proof for why $\|x\|:=\inf\{\lambda>0\mid\frac{x}{\lambda}\in B\}$ is a norm

I started studying functional analysis, a claim that was thought is the second lecture claims that: Let $X$ be a vector space, $B\subseteq X$ is convex, symmetric around $0$ and s.t ...
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1answer
313 views

Legendre Transformation of a Lagrangian in Classical Mechanics

I have some questions about the Legendre Transformation of a Lagrangian in Classical Mechanics to the Hamiltonian: We start with a Lagrangian $L(q,\dot{q})=\frac{\langle \dot{q} , \dot{q}\rangle }{2} ...
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1answer
90 views

Testing for Convexity for a function

Please any one can help figure out if this funcion is concave or convex, any help is greatly appriciated. Any links on how to test fo convexity for such a function is also greatly appriciated. I tried ...
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161 views

The convex function proof.

Can anyone help me prove convexity conditions? I know convexity conditions. The second derivative function is greater 0 first order convexity conditions. convex function conditions Because my ...
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1answer
110 views

Is $f(x,y) = x^\beta/y$ quasi-convex for positive $x,y$ for any real $\beta \geq 1$?

A multivariate function $f:{\mathbb R}^d \to {\mathbb R}$ is quasi-convex on a convex set $S \subset {\mathbb R}^d$ if $f(\lambda z + (1-\lambda)z') \leq \max\{f(z),f(z')\}$ for all $z,z' \in S$ and ...
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n-simplex in an intersection of n balls

Consider a $n$-simplex, $n \geq 2$ with vertices $x_i,i=1,...,n+1$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ ...
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243 views

How prove this inequality generalized from 1969 IMO problem 6

Let $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},w_{1},w_{2} $ are all positive numbers, and such $$x_{1}y_{1}z_{1}-w^3_{1}>0,\; \text{ and }\;x_{2}y_{2}z_{2}-w^3_{2}>0.$$ show that ...
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3answers
210 views

A secant inequality for convex functions

Suppose $f(0) =0 $ and $0<f''(x)<\infty (\forall$ $x>0)$, then $\frac{f(x)}{x}$ strictly increases as $x$ increases. I have shown that $f'(x)-\frac{f(x)}{x} = \frac{1}{2}xf''(c)$, for ...
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2answers
216 views

Convexity of log sum function

Is $f\left( x \right)=\log \left( \sum_i \beta_i e^{-\alpha_ix} \right)$ a convex function where $\beta_i,\alpha_i\in \mathbb{R}$?
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42 views

question about positive concave functions.

I am quoting a line from a text: The Laplace exponent $\Phi$ is concave and non-negative, the inequality $\Phi(\lambda)\leq k\Phi(\lambda/k)$ for all $\lambda>0$ and $k>1$ follows. Why does ...
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140 views

Local Extrema and Global Extrema

When we have a convex function we know that a local minimum is a global minimum, and similarly for a concave function. What are some other situations where finding local extrema can yield global ...
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60 views

Interpreting a theorem about convex sets

I'm studying about linear programming and I bumped into following theorem (I have added my questions into the image): So in 1st rectangle how did we obtain the resulting equation when $\alpha ...
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1answer
115 views

Help with a proof of a theorem about convex sets

I'm studying the following theorem: Theorem 1. Let $C$ be a convex set and let $\textbf{y}$ be a point exterior to the closure of $C$. Then there is a vector $\textbf{a}$ such that ...
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1answer
91 views

Equivalent definitions of piecewise affine convex function over a convex set of $\mathbb{R}^n$.

Let $C\subset\mathbb{R}^n$ be a convex set and $f:C\to\mathbb{R}$ a convex function. I want to show that the following are equivalent: $\mathrm{epi}(f)=\{(x,y)\in C\times\mathbb{R}\mid y\geq f(x)\}$ ...
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1answer
133 views

Convex functions and Hahn-Banach application

Let $Z$ be a convex subset of a real vector space, and $f:Z \to \mathbb{R}^m$ be such that every component $f_i:Z \to \mathbb{R}$ is a convex function. Let $S:\mathbb{R}^m \to \mathbb{R}$ be defined ...
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403 views

The unit ball in a Hilbert space

I have a request for any ideas to prove: If $H$ is a Hilbert space, then any unit vector is an extreme point of the unit ball of $H$. Every isometry is an extreme point of the unit ball of the ...
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Minimisation of Concave Function

$f\left(\mathbf{x}\right):\mathbb{R}_+^n\rightarrow\mathbb{R}_+$ is a concave monotonically increasing function to be minimised over the feasible region $\sum_{i=1}^n x_i=1$ and $x_i\geq ...
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236 views

closed epigraphs equivalence

Is there a way to prove that the epigraph of any real function $f$ is closed iff $f$ is lower semi-continuous without using limit superior or inferior?
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Is the function of two strictly concave functions also concave?

This may be a trivial question to most, but here we go: I have two strictly concave functions, say $f(x)$ and $g(x)$. From this can I say that a function of those two functions, $h[f(x), g(x)]$, is ...
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1answer
130 views

Graphics clipping: How can repeated half-space clipping fail?

Hi I am currently going through the past exam problems and I am stuck on this clipping problem. Could you give me some hint on how to solve it? If we clip a polygon to a window, it is inadequate ...
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332 views

when is the epigraph a convex cone?

The problem is from Stephen Boyd's textbook, which I couldn't solve. The question is "when is the epigraph of a function a convex cone?" The solution says that it is when the function is convex and ...
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1answer
416 views

A convex function with a Lipschitz continuous always has a strong convex conjugate function.

A smooth convex functions with $C^1$ has not always a Lipschitz continuous gradient. Please see the answer. If $F$ is convex and has a Lipschitz continuous gradient with modulus L, then $F^*$ is ...
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Separating hyperplane condition for complex vector spaces

I am only learning convex analysis properly now for the first time, and most of the references I am using only deal with topological vector spaces over $\mathbb{R}$. Is there any serious stumbling ...
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isoperimetric inequality using Fourier analysis

I'm trying to prove an isoperimetric inequality, but I have absolutely no idea how to go about it. let $\Gamma$ be a closed plane curve parametrized by $\gamma(t) = (x(t), y(t))$ on $[-\pi, \pi]$. ...
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What are the main uses of Convex Functions?

Up till now I have just learned that the concept of convexity in functions of one variable is used to complete the graphs of functions, meaning to locate points of inflexion and see if the graph is ...
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Is Positive Semidefinite matrix Same as Positive Number in Convex Optimisation?

Consider the optimisation problem expressed in a crude form $\max_{\mathbf{Q}}\sum w_ir_i$ where $w_i$ are constants, $r_i$ are concave functions of positive semidefinite matrix $\mathbf{Q}$ ...
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1answer
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Optimization problem interpretation

I posted a question in http://math.stackexchange.com/ and got a solution. But the solution is a bit hard for me to understand. The actual question is here : minimizing $\sum_{i=1}^n \max(|x_i - x|, ...
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Procedure to find generators of the dual cone

Page 11 of Fulton's "Toric Varieties" gives the following procedure for finding generators of the dual of a convex polyhedral cone in $\mathbb{R}^d$: For each set of $n-1$ independent vectors ...
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Number of supporting hyperplanes

I know that, for any convex set $S$, there is at least one supporting hyperplane at every point in $B$, the boundary of $S$. Also, there can be more than one supporting hyperplane at the same point in ...
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1answer
63 views

a function is approximated by a convex function

Let $g$ be a positive function on $(0,1)$ such that $g(x)\to\infty$ as $x\to 0$. Then, there exists a convex function $h$ on $(0,1)$ such that $h\leq g$ and $h(x)\to\infty$ as $x\to 0$. We can find ...
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Is alpha divergence a convex divergence measure?

Alpha divergence is defined as following : $$ D_\alpha(p||q) = \frac{1}{\alpha (1-\alpha)} \left( 1- \int _x p(x)^{\alpha} q(x)^{(1-\alpha)} dx \right) $$ if the distributions are restricted to ...
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A weak version of Markov-Kakutani fixed point theorem

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a commuting family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element ...
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223 views

Prove convexity of complicated rational function

Can anyone help me prove the convexity of this rational function? The man who proved the convexity of function used these facts. But I don't know this fact is correct or not. Here are the facts and ...
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1answer
213 views

Von Neumann's minimax theroem and Carathéodory's theorem

In J.F. Mertens(1986)(Paywall), there's a neat proof of a version of Von Neumann's minimax theroem. But I can't understand how Carathéodory's theorem is invoked. Suppose, in a two-person zero sum ...
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Checking convexity from outside

Is there any method or algorithm to determine convex (or non-convexity) property of a region from outside (perimeter) ? One way is plotting tangent line in each point of perimeter and discuss how ...
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252 views

Find the maximum convex area

My question is very similar to Plow's Question; but with this difference: How can I find the maximum convex area that can fit inside a non-convex region? For an example, consider this non-convex ...
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81 views

Subdifferential calculus

Let $\phi : H \to \mathbb{R}$ ($H$ is a vectorial space) be a convex function $\mathcal{C}^1$. I have the following inequality, for $\sigma \in H$ fixed, $$\forall \tau\in H, \ (\sigma - \tau \mid ...
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132 views

Supporting hyperplanes Theorem in Boyd's Convex Optimization

On page 51, the authors applied the separating hyperplane theorem to the sets ${x_0}$ and the interior of $C$ to prove the supporting hyperplanes theorem (assuming the interior of $C$ is nonempty). ...