# Questions tagged [convexity]

For questions involving the concept of convexity

513 questions
Filter by
Sorted by
Tagged with
113 views

### The “interior” of a convex set?

Consider any compact metric space $S$ which is a convex subset of a vector space. I am trying to see if the following definition is equivalent/close to some well-known definitions. Consider a convex ...
555 views

### Simple-looking problem with integrals

Let $f: [0,\infty) \rightarrow \mathbb{R}$ be a continuous function such that $f(0) = 0$. Is it true that if the integral $$\int_0^{\pi/2} \sin(\theta) f(\lambda \sin(\theta)) \, d\theta$$ is zero ...
55 views

### Positive scalar curvature on the double of a manifold (assuming mean convexity of the boundary)

This question is related to a previous one. Let $(M^n,g)$ be a compact Riemannian manifold with boundary. Assume it has positive scalar curvature and $\partial M$ is mean convex (positive mean ...
48 views

126 views

### Extremal bases in finite-dimensional Banach spaces

Definition. A basis $e_1,\dots,e_n$ for a Banach space $X$ is called extremal if there exists a point $s$ in the unit sphere $S_X=\{x\in X:\|x\|=1\}$ such that for every $i\in\{1,\dots,n\}$ the ...
135 views

### The set of boundary vectors of compact convex body has empty interior

Let $K$ be a compact convex body in the Euclidean space $\mathbb R^n$ and $\partial K$ be its topological boundary in $\mathbb R^n$. Definition. A vector $\mathbf v\in\mathbb R^n$ is called $K$-...
4k views

Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem. Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\... 0answers 48 views ### A standard name of a strongly extremal point of a convex set I need to name somehow points$x$of a bounded convex set$C$in a Banach space$X$such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at$x$has non-empty interior in the ... 0answers 145 views ### Sufficient condition for geodesic convexity/connectedness Let$(\Sigma,g)$be a connected smooth Riemannian manifold without boundary. By a minimizing geodesic I mean a geodesic whose length equals the distance between its endpoints. Let us consider the ... 0answers 46 views ### convexity of a function inside expected value Assume that$g(x)=\int_{y\in Y}h(y)f(x,y)dy$. I know the properties of function$g(\cdot)$, e.g. first- and second-order derivatives signs. Is there a way to drive the same properties i.e. convexity ... 0answers 36 views ### Stationary distributions of convex combination of stochastic matrices Consider two irreducible finite state Markov chains with transition matrices$A,B\in\mathbb{R}^{n\times n}$. Let$x$and$y$be the unique stationary distributions of$A$and$B$, respectively. Now ... 0answers 34 views ### Convex separation in CAT(0) spaces Are there any convex separation theorems for CAT(0) spaces? Particularly, can one separate two disjoint convex sets by a horosphere? In Riemannian manifolds, I have seen some results on convex ... 0answers 46 views ### Conjugate of composition in Bochner spaces Let$H$be a separable Hilbert space (of non-zero dimension), let$(\Omega,\Sigma,\mu)$be a finite measure space, and let$L^2(\mu;H)$be the Bochner-space$\mu$-integrable$H$-valued functions. ... 0answers 16 views ### Convexity and supermodular functions Suppose that the function F(x,y) is supermodular in (x,y). We know that this implies that x* =argmax F(x,y) is increasing in y. Under which conditions (if any) we have that x* is a convex function of ... 0answers 49 views ### Decomposition of Polyhedral - An example There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system $$\... 1answer 48 views ### Planar function inequality on parallelograms Let f be a function defined on the unit square R = [0,1]^2 \subseteq \mathbf{R}^2 which is convex and satisfies \frac{\partial{f}^2 }{\partial{x}\partial{y}} \leq 0. The last condition is ... 3answers 731 views ### Why the sequence of Bernstein polynomials of \sqrt x is increasing? Bernstein polynomials preserves nicely several global properties of the function to be approximated: if e.g. f:[0,1]\to\mathbb R is non-negative, or monotone, or convex; or if it has, say, non-... 0answers 46 views ### Second-order def of strong convexity wrt general norms A function f: R^n \rightarrow R is said to be C-strongly convex with respect to a norm \|\cdot\| if for all x,y and \lambda \in [0,1]$$f(\lambda x + (1-\lambda)) \le \lambda f(x) + (1-\lambda)... 1answer 72 views ### Sufficient conditions for the convexity of the discrete Fourier transforms Let$f : [0,2\pi] \to \mathbb{R}$be some function. Then the discrete Fourier transform of$f$when sampled at$2\pi i/Nis then given by $$X_n := \sum_{i=0}^{N-1}\cos\left(\frac{2\pi n i}{N}\right)... 1answer 101 views ### Convexity of discrete Fourier transform Let f : [0,2\pi] \to \mathbb{R} be a continuous convex function on (0,2\pi) which is singular about 0 and 2\pi but finite when evaluated at the boundaries. Assume also that f is symmetric ... 2answers 427 views ### Is every face exposed if all extreme points are exposed? Let C be a non-empty compact convex subset of {\mathbb R}^d such that every extreme point of C is an exposed point of C. Does it follow from this that every face of C is an exposed face? 1answer 130 views ### Existence of an asymptote for g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right) Working with the Slater's inequality (compagnion of Jensen's inequality) I find this statement : Let f(x) be a continuous,twice differentiable function ,convex or concave and non constant on (0,\... 1answer 37 views ### Is this relation between planar convex hulls and heaviest cliques true? If P is a set of n points in the euclidean plane whose convex hull \operatorname{CH}(P) has h corners, and Q\subset P has m\le\lfloor\frac{h}{2}\rfloor points and maximal sum of pairwise ... 0answers 304 views ### Is there yet an example of a non-negative convex polynomial that cannot be written as a sum-of-squares? I have read that it remains an open question, whether an example can be constructed of a non-negative convex polynomial that cannot be written as a sum-of-squares. My reading includes the following ... 0answers 60 views ### Characterization of gradient of convex conjugate Let X be a separable Hilbert space. Is there a known characterization of all functions F:X\rightarrow X for which there exists a proper (not identically \infty), lower semi-continuous convex ... 0answers 15 views ### Can planar convexity be decided with matching This question actually aims at generalizing the notion of a finite set of points in the euclidean plane being in convex configuration to complete , undirected simple graphs with arbitrary edge-weights.... 1answer 232 views ### Closedness of linear image of positive L1 functions Let \mathcal X be the Banach space of L^1 functions on some probability space, \mathcal Y be some other Banach space, T:\mathcal X\to \mathcal Y be some surjective continuous linear map, \... 0answers 64 views ### Sparse signal recovery (nonlinear case) Let K \subset \mathbb{R}^n, it may be that K is "very thin" (e.g. K is a k-dimensional affine subset of \mathbb{R}^n, with k \ll n). I'm interested in the case where K is ... 0answers 34 views ### Self-duality of cones associated with elementary symmetric polynomials Let n\ge3 be an integer, and denote \sigma_1,\ldots,\sigma_n the elementary symmetric polynomials in n indeterminates:$$\sigma_1(X)=X_1+\cdots+X_n,\quad\ldots\quad,\sigma_n(X)=X_1\cdots X_n.... 0answers 73 views ### About the definition of lineal convexity I have been trying to understand the definition of lineal convexity. I am reading the article Duality of functions defined in lineally convex sets by Christer O. Kiselman. For a set A\subset \mathbb{... 1answer 102 views ### Expectation of random matrix Assume Q is a positive definite random matrix such that 0 < \lambda_{\min}(Q)....\leq \lambda_{\max}(Q) \leq 1 holds. I want to show that \begin{align} E\left[\frac{\lambda_{\min}(Q)}{\lambda_{\... 1answer 143 views ### Projection of convex set onto a convex set [closed] Can projection of convex sets onto convex sets be non-convex yet connected? If so is there any necessary and sufficient conditions? Can projection of n dimensional convex sets in \mathbb R^{n'} ... 2answers 190 views ### A question about asymptotic affinity and strict convexity with unbounded means Let F:[0,\infty) \to [0,\infty) be a C^1 strictly convex function. Let \lambda_n \in [0,1],a_n\le c<b_n \in [0,\infty) satisfy \lambda_n a_n +(1-\lambda_n)b_n=c_n \tag{1}$$and assume that ... 1answer 154 views ### Does strict convexity plus asymptotic affinity imply bounded mean? I am not sure if this is exactly research-level, but I am struggling to find a proof for the following claim: Let F:[0,\infty) \to [0,\infty) be a C^2 strictly convex function. Let \lambda_n \in [... 1answer 269 views ### Closed convex hull in infinite dimensions vs. continuous convex combinations tl;dr: When is the closed convex hull of a set K equal to the set of "continuous" convex combinations of K? I am essentially asking for the most general, infinite-dimensional analogue of ... 1answer 180 views ### Is this (\Bbb R^{n \times n})^n \to \Bbb R function convex? Let W := (W_1, W_2,\dots, W_n), where W_i \in \Bbb R^{n \times n}. Let x be a constant vector. Is the following function convex?$$f(W) := x^TW_1^TW_2^T \cdots W_n^TW_n \cdots W_2W_1x $$0answers 119 views ### Log-concavity inequality Let x,y, and t be fixed real numbers, 1<x<y, 0<t<1. Does the following inequality hold for some c$$\frac{\log{(tx+(1-t)y)}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{(sw+(1-s)z)}}{... 2answers 395 views ### Is a function of several variables convex near a local minimum when the derivatives are non-degenerate? This is a cross-post. LetU \subseteq \mathbb R^n$be an open subset, and let$f:U \to \mathbb R$be smooth. Suppose that$x \in U$is a strict local minimum point of$f$. Let$df^k(x):(\mathbb R^n)^...
Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Assume (for simplicity) that $M$ is compact. Let $M$ be locally geodesically convex, i.e. any shortest path in $M$ ...