Tagged Questions

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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On a Banach space $X$, is the functional $x \mapsto \frac{1}{p}\|x\|^p$ convex?

Let $X$ be a Banach space. Let $p > 1$ and, consider the functional $X \to \mathbb{C}$ given by: $$x \mapsto \frac{1}{p}\|x\|^p$$ I would like the know if the above functional is convex. That ...
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Prove that a Hilbert space is convex of power type $2$

Let $X$ be a Banach space. For $\epsilon \in (0,2]$, define: $$\delta_X(\epsilon) = \inf_{x,y \in X}\{1 - \|\frac{1}{2}(x + y)\| : \|x\| = \|y\| = 1, \|x-y\| \ge \epsilon\}.$$ Then we say that $X$ ...
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Are strictly convex functions with positive second derivatives on compact domains strongly convex?

Claim: Let $\chi$ be a compact set. If $f''(x)>0$ for all $x\in\chi$, then $f$ is strongly convex. This seems to be true, intuitively, as I can't think of a counterexample. All of the examples I'...
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Knuth's Sandwich Theorem: requesting proof clarification

The question is about F6 of Section 8 ("Elementary facts about cones") in Donald Knuth's Sandwich Theorem (http://arxiv.org/pdf/math/9312214.pdf). He claims to prove $(A \cap B)^* = A^* + B^*$ when $A$...
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Is every such function convex or concave?

I was pondering convex functions today, and the following questions naturally posed themselves. Call a function $f : \mathbb{R} \rightarrow \mathbb{R}$ $2$-limited iff for all $m,c \in \mathbb{R},$ ...
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Convexity of functions

I'm trying to read up on convex/concave functions (is that the same as concave up, concave down?) If I were asked to prove a convexity of a function, what are the general steps to follow? (So far I ...
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Convex surface on which any two points $a,b$ can be joined by a curve of length $(\pi/2-\epsilon)|a-b|$

I am trying to solve an exercise on page 13 of the book Metric structures on Riemannian and non-Riemannian spaces by Gromov. Construct a closed, convex surface $X$ in $\mathbb R^3$ such that any ...
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Quasi-Concavity and Quasi-Convexity

My book states that: $f$ is a quasiconcave function on $U$ if for all $x,y \in U$ and $t \in [0,1]$: $f(x) \geq f(y) \implies f(tx + (1 - t)y) \geq f(y)$ $f$ is a quasiconvex function on $U$ ...
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separating hyperplane theorem proof

I need help understanding the last part of this proof, the lemma they are refering too is just a lemma about convex sets that shows us that p is unique: I do not see how H* separates C and z. The ...
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$F(x) = f(x) + g(x) + h(x)$, where h(x) is strongly convex , is also strongly convex

$\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ $\newcommand{\Tr}{\operatorname{Tr}}$ Suppose $g: \mathbb{R}^n \rightarrow \mathbb{R}$ is a continuous convex ...
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Prove that the set $C = \{x \in\Bbb R : ax\le b\}$ is convex

Prove that if a and b are real numbers, then the set $C = \{x \in\Bbb R : ax\le b\}$ is a convex set. My solution so far: To show that a set $C$ is convex it needs to be shown that for for every ...
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Convexity on a direction

In "Ming-Jun Lai, Larry L. Schumaker. Spline Functions on Triangulations. Cambridge University Press, 2007, p.72." we have: A function $f$ defined on a triangle $T$ is said to be convex in the ...
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Prove vertices of a simplex are affinely independent

I'm given that the definition of a simplex $T$ is $x \in\mathbb R^n$ such that $x$ satisfies $n+1$ linear inequalities: $(u_k, x) \lt c_k$ for $k = 1,\ldots,n+1$ (i.e. $T$ is the intersection of $n+1$ ...
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What is a $0$-sublevel set?

I read the notes of S. Boyd, and am confused about the following: $f_0(x)$ is quasiconvex. I am confused about the latter one particularly. What does it mean? Thanks!
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Non-trivial compatibility which makes convex functions continuous on $\Bbb R$

Here are the definitions: Let $X$ be a set. Another set $\mathcal C\subseteq \mathcal P(X)$ is called a convexity over $X$ if $\varnothing, X\in\mathcal C$ $\mathcal C$ is closed under arbitrary ...
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Why is it a borel set on the boundary of the unit ball of $E^n$?

Given $C$ convex body (compact convex set with non-empty interior points) in $E^n$ symmetric about the origin and containing the unit ball. Let $A(r)$ denote ,for every real $r >1$, the subset of ...
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what is the meaning of asphericity of convex set in a linear normed space?

I am trying to understand the definition of spherical to within $\epsilon$ for a convex body in a linear normed space, as given in Aryeh Dvoretzky's paper [1] (section 2, page 203): A convex set $C$...
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