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Questions tagged [convex-analysis]

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.

Convex analysis is the study of properties of convex sets and convex functions.

Formally, a set $S$ is convex if, for all $x, y \in S$ and $t \in [0,1]$, $tx + (1-t)y \in S$. Intuitively, this says that for any two points in $S$, the line segment connecting those points is also in $S$.

Formally, a function $f: S \to \mathbb{R}$ defined on a convex set $S$ is convex, if for all $x,y \in S$ and $t \in [0,1]$, $f(tx + (1-t)y) \leq t f(x) + (1-t) f(y)$. Intuitively, this says that, for any two points on the graph of $f$, the line segment connecting those points lies above the graph of $f$.

Some of the more important results in convex analysis include Carathéodory's Theorem, Jensen's Inequality, Minkowski's Theorem, and the supporting hyperplane theorem.

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$\mu$-strongly convex is strictly convex

From this definition, how to prove $\mu$-strongly convex is strictly convex? I know there are some similar questions in the website, but they use the definition containing differentiable. But that is not necessary. The defition of $\mu$-strongly…
Jonathen
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How to show that a convex set is bounded

Let A be a convex subset of $$R^{2}$$ containing the origin and possessing the following property: given any constants $$\alpha_{1}, \alpha_2\in R$$ such that $$\vert \alpha_1 \vert + \vert \alpha_2 \vert > 0,$$ the subset $$\{ (t \alpha_1,t\alpha_2…
JHY
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Proving an entire family of convex sets has non empty intersection given that every denumerable subfamily has non empty intersection

This is a question from Brank Grunbaum's "Convex Polytopes": Let $\{K_v\} $ be a family of convex sets in $\mathbb R^d $ such that every denumerable subfamily has non empty intersection, then $\cap_v K_v \neq \emptyset$ The issue here(and probably…
giorgioh
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Conditions under which the Minkowski surface equals the surface area

The Minkowski surface of a convex, compact set $K$ can be defined as (here, I am using Kurt Leichtweiß: Konvexe Mengen, Definition 16.2) $$O(K):=\lim_{\epsilon\rightarrow 0+} \frac{V(K_\epsilon)-V(K)}{\epsilon},$$ where…
Ina
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Does there exist a finite,nowhere continuous convex function that is not bounded in any interval

the statement comes from Natanson's real analysis.Firstly,he shows a theorem which says that if $f(x)$ is a bounded low-convex function in $[a,b]$,then $f(x)$ is continuous in $(a,b)$,then he appends to say that "the condition bounded is very…
math
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Logarithmically Convex Functions Characterization

I need help proving that if $g(x)=e^{\langle a,x\rangle }*f(x)$ is convex on $\mathbb{R}^n$ for all $a \in \mathbb{R}^n$, then $f$ is logarithmically convex. One thing I noticed was that if $g$ is log-convex itself, then $f$ is log-convex since…
Taurseng
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Why is a slab convex?

A set of the form $$\left\{x \in \Bbb R^n\mid \alpha \leq a^T x \leq \beta \right\}$$ is called a slab. A slab is an intersection of two halfspaces. Hence, it is a convex set. Can someone explain why a slab is an intersection of two halfspaces? I…
XYZ
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How to prove $||X||^p,$ $p>1$ is strictly convex?

How to prove $||X||^p,$ $p>1$ is strictly convex? Here $X\in\mathbb{R}^d$, and $||x||$ is the $l_2$-norm. I know how to prove $||X||^p$, $p\geq1$ is convex, for example, using Holder. But how to prove the strict convexity?
Tan
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Inverse of $\sum^n_{i=1} 1/x_i$ convex?

Is this function convex? $$f(x)=\left(\sum^n_{i=1} 1/x_i \right)^{-1}$$ where $\operatorname{dom} f =\mathbb{R}^n_{++}$. I having a hard time coming up with a clean Hessian from the partial derivatives.
darisoy
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On a characterization of continuous convex functions

I'm trying to prove the following know result: Theorem Let $I = (a,b)$ be an interval, and let $\phi \colon I \to \mathbb{R} $ be a continuos function. Then, $\phi$ is convex in $I$ if and only if $$(*) \quad \phi(x) \leq…
fcz
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Linear Difference Equations

Let $g_1, g_2 \in \mathbb{R}^{[0,5]}$ be defined by $g_1(x) = x$ and $g_2(x) = 1-x$ for each $x \in [0,5]$. Find a second-order homogeneous linear difference equation on $[0,3]$ such that {$g_1, g_2 $} forms a fundamental set of solutions to that…
DreamLighter
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Relationship between $f''$ and "$f$ is strongly convex".

Suppose $f$ is strongly convex and twice differentiable on some interval $I$: (1) $\forall x,y \in I: \forall t \in (0,1) \implies f(tx + (1-t)y)
user838035
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Is Lower Envelope of Compact and Convex Set always continuous?

From a picture in $\mathbb{R}^2$, say a square $[0, 1]\times [0, 1]$, the lower envelope of the square is $[0, 1]\times 0$ and can be seen as a continuous function over $[0, 1]$. I wonder if there exists a general theorem proving this intuition or…
Chasel Weng
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Show $f(x_1,x_2) = \frac{1}{x_1x_2}$ is convex for $(x_1,x_2) \in \mathbb{R}^2_{++}$

From Problem 3.16(c) of Boyd & Vandenberghe's Convex Optimization: Determine if $f(x_1,x_2) = \frac{1}{x_1x_2}$ is convex, quasiconvex, concave, or quasiconcave on $\mathbb{R}^2_{++}$. From this post Determining whether $\alpha$ sublevel sets are…
user594147
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$|x|^{3/2}$ strongly convex?

$$f(x) = |x|^{3/2}, x \in \mathbb{R}$$ If you take the 2nd derivative you get $$ f''(x) = \frac{3}{4\sqrt{|x|}}$$ If you go by the requirement that $f$ is strongly convex when $$f''(x) \geq m \gt 0$$ Then $f$ is strongly convex. However, if you go…
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