# Questions tagged [convexity]

For questions involving the concept of convexity

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### Graduate level convexity - Intersection of an r-polytope with a hyperplane is an r-1 polytope

I am trying to follow Roger Webster's Convexity 's proof of Euler's celebrated result on the relationship between the number of faces of a polytope. An image of the proof is here. In the course of the ...
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### Is this function concave? If it is, can we show it in a theoretical way?

Suppose we have a function: \begin{equation} \begin{aligned} f(x_1,x_2,\cdots,x_n)&=\sum\limits_{i=0}^n\frac{(x_i e^{-x_i}-x_{i+1}e^{-x_{i+1}})^2}{e^{-x_i}-e^{-x_{i+1}}}\\ &=\frac{(x_0 e^{-x_0}...
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### Lovasz extension and underlying matroid

Let $F \colon 2^V \rightarrow \mathbf{R}$ be a set function for some ground set $V = \{ 1, \dots, n\}$. The domain of $F$ is $\{ 0,1\}^n$ , under the usual identification of a set $S$ with its ...
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### Injectivity of a convex combination of squares $A^*A$ in $\ell^\infty$

Consider two operators $A,B: \ell^p \to \ell^p$ ([defined and]bounded for all $p \in [1,\infty]$) as well as their adjoints $A^*,B^*:\ell^p \to \ell^p$. Assume $A^*A$ and $B^*B$ have trivial kernel ...
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### A convexity question

Let $Q=[0,1]\times[0,1]$ and let $a$ be a positive smooth function on $Q$. Does there exist a smooth positive function $u$ On $Q$ such that there holds $$\frac{\partial^2}{\partial x_1^2}u <0$$ ...
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### Regarding definition of convex cone and apex

I have been reading about star shaped sets and support cones from this article. I am wondering about the definition of the cone as to why is it defined this way? Why is it $C-a$? I am familiar with ...
1 vote
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### minimum eigenvalue interpolation

Suppose we have two symmetric positive definite matrices $A,B$ (not simultaneously diagonalizable). How can I find a matrix function $f(t), t\in [0,1]$, such that $f(0)=A, f(1)=B$, and the minimum ...
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### Is $0$ a member of the following special kind of a convex compact set?

Let $(V, \lVert \cdot \rVert)$ be a normed space. Let us consider the set $C = [-1,1]^{\dim V}$. The boundary of this set consists of closed subsets $B_i$ (indexed by some set $I$) of affine ...
1 vote
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### Minimum bounding rectangle of symmetric convex bodies in the plane : is the ball the worst case

The minimal rectangle containing the euclidean ball in the plane is the standard cube $B_\infty = [-1;1]^2$. I would like to know if the euclidean ball is the worst symmetric convex body to be ...
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Let us recall that $\mathfrak b$ is the smallest cardinality of a subset of $\omega^\omega$, which cannot be covered by countably many compact subsets of $\omega^\omega$. The definition of $\mathfrak ... 2 votes 0 answers 50 views ### Does absolute retract imply convex structure? In the theory of selection, it is known that any compact absolute retract (AR) carries a convexity structure defined by E. Michael. It is also known that a convex structure developed by Van de Vel ... 3 votes 0 answers 136 views ### Reference request for convex geometry? I am looking for a reference for an elementary convex geometry. In Appendix A (page 1810) of this paper by Green and Tao, they cover some basic results from elementary convex geometry. The results ... 17 votes 1 answer 233 views ### Functions of$\mathbb{R}^d$preserving convexity of sets Consider a function$f : \mathbb{R}^d \to \mathbb{R}^d$, with$d\geq 2$, such that:$f$is injective, For any convex set$A$of$\mathbb{R}^d$,$f(A)$is also convex. What can we say about$f$? In ... 5 votes 2 answers 141 views ### Which convex subsets of a normed space are intersections of balls? Let$(V, \lVert \cdot \rVert)$be a normed space. For any$A \subseteq V$, let$O(A)$be the intersection of all closed balls containing$A$, or more precisely, let$O \colon 2^V \to 2^V$be defined ... 3 votes 0 answers 65 views ### Projection onto level set of convex functional Fix a probability space$(\Omega,\mathcal{F},\mathbb{P})$and let$F:L^2_{\mathbb{P}}(\mathcal{F})\rightarrow (-\infty,\infty]$be bounded-blow, convex, lower semi-continuous, and not identically ... 1 vote 1 answer 77 views ### Convexification of difference of convex functions I am looking for a reference/a hint to the following problem: We are given$f_1(x),f_2(x)$convex functions (say, on$\mathbb R^d$) such that$f_1(x) \to\infty$for$\|x\|\to\infty$. Also there is an$...
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$f \in C^2([0,1])$ with $f''$ convex and $f(0) = f'(0) = f''(0) = 0$. Is it true that : $f''(1)+6f(1)\geq 4f'(1)$ ? Source: AoPS
1 vote
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### Maximal geodesically convex function interpolating three points on the hyperbolic plane

Crossposted on MSE: https://math.stackexchange.com/questions/4282998/maximal-geodesically-convex-function-interpolating-three-points-on-the-hyperboli Let $M$ be a two-dimensional Hadamard manifold. ...
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### Worst convex compact set for translational packings of $\mathbb R^d$

A (translational) packing of a convex compact subset (with non-empty interior) $\mathcal C$ of $\mathbb R^d$ is a union of translated non-overlapping (but perhaps touching) copies of $\mathcal C$. The ...
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### Log-concavity of the modified Bessel function of a second kind

I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified ...
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### Inscribed square and convexity

Let $b(X)$ be the boundary of any $X$ subset of the plane. Does there exist $A,B$ convex compact sets of the plane, such that $C:=A\setminus B$ is simply connected and not empty, and such that ...