Questions tagged [support-function]

For questions about the support function, a tool in convex geometry to represent sets via their support planes.

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The support of the Schwarz rearrangement of a compactly supported function

Let $E \subset \mathbb{R}^N$ a measurable set of finite measure. We define $E^\ast$, the symmetric rearrangement of $E$ to be $$ E^\ast := B_R(0), R = \left(\frac{|E|}{|B|}\right)^{\frac{1}{N}}. $$ ...
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Support of function

As reported in https://en.wikipedia.org/wiki/Support_(mathematics), for a function $f:X \rightarrow \mathbb{R}$ we can define some notion of $supp(f)$, in particular: If $X$ is only a set, we define ...
Manuel Bonanno's user avatar
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Using Urysohn’s Theorem to prove there exist a continuous function with a given bounded $E$.

Let E be a bounded set in $\mathbb{R}^d.$ Then there exist a closed ball $B(0,K),$ $K>0$ such that $E \subset B(0,K).$ Show that there exist a continuous $f$ on $\mathbb{R}^d$ and $0\leq f(x) \leq ...
Lilili123's user avatar
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What is the connection between the gradient acting on the support function in Euclidean space and on a sphere?

The support function defined on the unit sphere can be locally represented as a support function defined in the entire space. This can be achieved by using a mapping \begin{align} (x_1,\cdots,x_n)&...
Serge's user avatar
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For $f\in\mathcal S'$, if $\varphi\in\mathcal D$ vanishes on a neighborhood of supp$f$, then supp$f\cap$supp$\varphi=\emptyset ?$

Let $f\in\mathcal S'(\mathbb R^n)$ and $\varphi\in\mathcal D(\mathbb R^n)$. Suppose $\varphi(x)=0$ for $x\in U$, where $U$ is a neighborhood of supp$f$. Then, does supp$f\cap$ supp$\varphi=\emptyset$ ...
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Supporting hyperplane through vector projection onto convex set

I am trying to proof the following property. Let $F_i: \mathbb{R}^m \rightarrow \mathbb{R}$ be a $c_1$ convex function $\forall i \in I$; $C = \{ x \in \mathbb{R}^m : F_i(x) \leqslant 0 \quad \forall ...
Matheus Diógenes Andrade's user avatar
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Can we conclude that $g \in C_c (V)\ $?

Let $U \subseteq \mathbb R^n$ be open and $f \in C_c (U).$ Let $V = U \cap B (0,1)$ be a non-empty open subset of $\mathbb R^n$ contained in $U.$ Can we conclude that $f \in C_c (V)\ $? The support of ...
RKC's user avatar
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Given the support function of a convex set $C\subseteq \mathbb{R}^{2n}$, compute $\sup \left \{c'y: (x,y) \in C\right\}$ as a function of $x$

Suppose $C \subseteq \mathbb{R}^{2n}$ is a closed, bounded, convex set, with support function $h: \mathbb{R}^{2n} \rightarrow \mathbb{R}$, defined as $$h(c_1, c_2) := \sup \{c_1'x + c_2'y : (x,y) \in ...
jackson5's user avatar
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support in complete manifolds

Let $(M,g)$ be a complete Riemannian manifold . Let $f\in C^\infty(\mathbb{R})$ such that $f\equiv1$ on $(-\infty,0]$ and $f\equiv0$ on $[1,\infty)$ . Let $x_0\in M$ be fixed and $d_g(x,x_0)$ be the ...
am_11235...'s user avatar
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Is it always possible to construct a distribution with full support?

Does any (arbitrary) measurable topological space admit a probability distribution with full support? If not, what is a counterexample?
Oliv's user avatar
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Determine the support of a gaussian vector

We consider $(X,Y,Z)$ a gaussian with mean $(1,2,1)$ and with covariance matrix: $$ \begin{pmatrix} 1 & -2 & 1\\ -2 & 5 & -1\\ 1 & -1 & 2 \end{pmatrix} $$ I am asked to ...
Pierre's user avatar
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If $K \subset \Bbb R^d$ is not bounded, is $h_K(x) = \sup_{y\in K} \langle x,y\rangle$ not Lipschitz?

Question: If $K \subset \Bbb R^d$ is not bounded, is $h_K(x) = \sup_{y\in K} \langle x,y\rangle$ not Lipschitz? Background. I came across the following proposition. If $K\subset\Bbb R^d$ is a ...
stoic-santiago's user avatar
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Can we say that $|\operatorname{supp}u\cap \{u=0\}|=0$?

This question intuitively sounds obvious at first glance. Surprisingly I am unable to provide a correct argument. Here is the matter we consider a measurable function $u\Bbb R^d\to\Bbb R$. Then the ...
Guy Fsone's user avatar
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The support function in ODE-PDE comparison principle of Ricci flow

Recent days, I want to understand the ODE-PDE comparison principle of Ricci flow which is origin from the Hamilton's paper Four-manifolds with positive curvature operator. But the Hamilton's paper ...
Enhao Lan's user avatar
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How dos a random variable $X$ is connected with the set over which is uniformly distributed?

I have a silly question, tough not obvious to me. Most times we see in probability theory many of the definitions or theorems start with the following quotation: ``Let $X$ be a random variable that is ...
Hunger Learn's user avatar
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Support for Cauchy’s principal value of $1/x$

I'm trying to find support for Cauchy’s principal value of $1/x$ given by $$\left\langle PV(1/x), \phi \right\rangle = PV \int \frac{1}{x}\phi(x)\mathrm{d}x \equiv \lim_{\epsilon \downarrow 0} \int _{|...
Liii's user avatar
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Conic representation of convex hull of unit ball and point

Let $S = \mathrm{conv}(B \cup p)$ where $B = \{x \in \mathbb{R}^2 | x^Tx \leq 1 \}$ and $p = (-2,0)$ Can this set be represented in conic form $Ax \preceq_{\mathcal{K}} b$ where $\mathcal{K}$ is $\...
James's user avatar
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Regarding Sup Norm of Function in $C_c(X)$

In the post (https://math.stackexchange.com/a/1729440/987565), the author states that $$\left\Vert \tilde{f}-f\right\Vert _{\sup}\leq\varepsilon$$ with $\tilde{f} = gf$, $f \in C_0(x)$, $g \in C_c(x)$...
user40102's user avatar
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Support of a convolution when extending $u\in W^{1,p}_0$ by zero (Brezis Ch 9)

(iii) $\Rightarrow$ (i). One can always assume that $\Omega$ is bounded (if not, consider the sequence $\left.\left(\zeta_{n} u\right)\right)$. By local charts and partition of unity this is reduced ...
Domenico Vuono's user avatar
3 votes
1 answer
532 views

Why is the support function of a convex cone the indicator function of its polar cone?

Let $K ⊆ \mathbb{R}^{d}$ be a non-empty, closed, convex cone. Consider the support function $\sigma_K(x):= \sup_{y \in K} \langle x, y \rangle$. This function describes the (signed) distances of ...
Pazu's user avatar
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Why $|g| \leq \|g\|_{\infty}$? Help with the notion of essential supp

I got stuck in the definition of the essential support of $f$. I basically want to prove Hölder's inequality for the case $p=1$ and $q= \infty$. The proof is based on the fact that $$|g| \leq \|g\|_{\...
Frederick Manfred's user avatar
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1 answer
137 views

Does it holds that the $L^{\infty}$ norm of the support function of a convex body is minimal on balls with the same volume?

I was wondering if the following inequality holds. Let $K$ be a convex body of $\mathbb{R}^n$ and let us denote by $h_K$ its support function, defined as, for $x\in\mathbb{R}^n$ $$ h_K(x)={\max}\{x\...
GGG's user avatar
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partition of unity \ multiplication in sobolev?

*let $\Omega$=$\omega_1$ × $\omega_2$ q1: why for any open set $ω'_1$ satisfying $ω'_1$ ⊂⊂ $ω_1$ , we can find an other open set $ω''_1$ such that $ω'_1$ ⊂⊂ $ω''_1$ ⊂⊂ $ω_1$ and a smooth function ρ ...
topb's user avatar
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How to recover a closed convex set from its support function?

Let $A\subset\mathbb R^n$ closed and convex. Its support function is defined as $h_A(u)=\sup_{x\in A}\langle x,u\rangle$, for every $u\in \mathbb R^n$. Being $A$ closed, there must be, for each $u$, ...
glS's user avatar
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Compact Support for a PDE

Suppose that $u$ is a $C^2$ solution to the wave equation in $\mathbb{R}^n\times\mathbb{R}$ Show that if $u(\cdot,0)$ and $u_t(\cdot,0)$ have compact support in $\mathbb{R}^n$, then $u(\cdot,t)$ has ...
ymmas's user avatar
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Support of a convolution: where is compactness used

$\def\supp{\operatorname{supp}}$ I've found some questions about this, but couldn't find an answer that had what I needed. I'm asked to prove that if $f,g \in L^1(\mathbb{R}^d)$ and $\supp(f)$ is ...
Silkking's user avatar
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Embedding of $S^n$ to the smooth strictly convex region in $R^{n+1}$

Assuming $D\subset R^{n+1}$ is smooth and strictly convex region, one can construct an embedding $\varphi$ of $S^n$ to the boundary $\partial D$ of $D$ which associates to each direction $z\in S^n$ ...
Enhao Lan's user avatar
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Are affine functions compactly-supported only when they are zero

Question: If $f$ is an affine map from $\mathbb{R}^n$ to $\mathbb{R}$ then must $f$ be supported on some affine subspace of positive dimension if and only if $f\neq 0$? So in particular $f$ is never ...
ABIM's user avatar
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Why $C\mapsto s(x^*,C)$ is affine function?

Let $X$ be a separable Banach space, the associated dual space is denoted by $X^*$ and the usual duality between $X$ and $X^*$ by $\langle , \rangle$. For $C$ nonempty weakly compact convex subsets of ...
Karim KHAN's user avatar
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1 answer
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Why does the support function equal the distance between $0$ and the supporting hyperplane?

I am currently revising the material in Luenberger's classic "optimization by Vector Space Methods" for an upcoming oral exam, and find myself stuck in section 5.13. I am stuck on the following ...
sortofamathematician's user avatar
1 vote
1 answer
61 views

Find the support function of $\{x\in\mathbb{R}^{n}|x^{T}Qx+2b^{T}x+c\leq0\}$

I need to find the support function of $S=\{x\in\mathbb{R}^{n}|x^{T}Qx+2b^{T}x+c\leq0\}$ where we assume Q is symmetric and definitely positive and we know that $S\neq\phi$. I thought using KKT method ...
user213158's user avatar
1 vote
1 answer
276 views

Is the domain of the support function of a subset of a simplex the unit sphere?

Consider the support function of a convex set $A\in \mathbb{R}^n$, defined as the map $$ x\in \mathbb{R}^n \mapsto \sup_{a\in A}x'a $$ Suppose now that $A$ is a subset of the $(n-1)$-dimensional ...
Star's user avatar
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1 vote
1 answer
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Show that is $K,L$ are convex bodies then $h_{\text{conv}(K\cup L)}=\max\{h_K,h_L\}$

Show that if $K$ and $L$ are convex bodies, then: $$h_{conv(K \bigcup L)}=max\{h_K, h_L\}$$ First doubt is: $h$ should be the support function, defined as: $h_C(u)=\sup\ \{ u \cdot y:y \in C \} \...
Valentina Sau's user avatar
1 vote
1 answer
691 views

How to find the explicit support function of convex set?

Let $S=A\cup B$ where $A=\{(x_1,x_2):x_1<0,x_1^2+x_2^2\le4\}$ and $B=\{(x_1,x_2):x_1\ge0,-2\le x_2 \le2\}. $ Find the support function. The support function is defined as $f(y)=\text{sup} \{y^tx:x\...
user441848's user avatar
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Retrival of convex hull by means of support function

My question may be driven by luck of appropriate knowledge. So I'm sorry if I ask something which maybe obvious. Generally support function are defined as $s_{A}(x) = \sup\{x\cdot a| a\in A\}$, for $...
kolobokish's user avatar
3 votes
2 answers
2k views

What is the support function of an ellipse?

I defined the support function $h_A:R^n→R$ of a non-empty closed convex set $A\subseteq \mathbb{R}^n$ as $$h_A(x)= \sup\left\{x⋅a |\ a \in A\right\}$$ Everything I know about this topic I found it. I ...
J.Doe's user avatar
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1 vote
1 answer
206 views

Prove that the support function satisfies $\sigma_{A+B}=\sigma_A+\sigma_B$ for $A,B$ compact convex sets

Let $A, B, C$ be compact convex sets in $\Bbb R^n$ such that $A + C = B + C$. The purpose of this problem is to prove that $A = B$. Define the support function $$\sigma _A (x) := \max\{\langle x, u\...
Paris J's user avatar
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2 answers
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Compute the support function, $\sigma_{C}$, when $C$ is a subspace.

Recall that in convex analysis the support function is the conjugate of the indicator function and is defined to be $\sigma_{C}(x)=\sup_{v\in C} \langle v,x\rangle$. Compute the support function, $\...
Jeremy's user avatar
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2 votes
1 answer
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How to retrieve the domain $S=\{(x,y): 0\le y\le x-1\}$ from its support function?

First the definition of the support function of set $S$: Let $S$ be a nonempty convex set. The support function $h$ of $S$ is the real-valued function defined by $$h(x)=\sup_{s\in S}\langle s,x\...
gelfand's user avatar
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1 vote
3 answers
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Prove that the support function of an arbitrary set $A\subset\mathbb R^n$ is convex

Given an arbitrary set $A ⊂ \mathbb{R}^n$ , the support function associated with the set $A$ $ σ_A : \mathbb{R}^n \to \mathbb{R} ∪ \{+\infty\}$ is defined as $\sigma_A(x):= \sup_{z \in A} \langle x,...
diabloescobar's user avatar
21 votes
1 answer
4k views

What functions are support functions of convex sets

Given a closed, convex, non-empty set $K\subseteq\mathbb{R}^n$ the support function $h_K:\mathbb{R}^n\to (-\infty,\infty]$ is defined as $$h_K(y) := \sup_{x\in K} \langle x,y\rangle$$ It is easy to ...
Johannes Hahn's user avatar
1 vote
2 answers
1k views

Prove that the support function of a closed non-empty set is convex [duplicate]

Let $C$ be a closed non-empty set, but not necessarily convex. The support function of $C$ is given by $$S(z) = \sup_{c \in C} \langle z,c\rangle. $$ Prove that this is a convex function. Attempt ...
Lemon's user avatar
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7 votes
1 answer
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Can we infer from the support function of a convex domain $Q$ when the boundary $\partial Q$ admits nonvanishing curvature?

Let $Q$ be a compact, convex domain in the plane, with smooth boundary $\partial Q$. We further assume that the origin is contained in $Q$. For a concrete example, let's take an ellipse $\frac{x^2}{a^...
Pengfei's user avatar
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1 vote
2 answers
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What is the set of $x$ achieving the maximum in the definition of the support function $h_B(u)=\sup_{x\in B}\langle u,x\rangle$?

Suppose that $B$ is a convex body (compact closed with nonempty interior), and let $$h_B(u) = \sup_{x \in B} \langle u,x\rangle$$ be its support function. Is there a nice description of the set $E :=\{...
passerby51's user avatar
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19 votes
1 answer
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What is the support function $h_K(x)\equiv \sup_{z \in K} \langle z, x \rangle$ of a set $K$?

I want to ask that, what is a support function intuitively. It is defined as: $$\sup_{z \in K} \langle z, x \rangle$$ where $z \in K$, $K$ is a nonempty set. In this formulation, $\langle \cdot, \...
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