# Tagged Questions

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### Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $\lambda > 0$) and the "...
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### Is Ryser's conjecture on permanent minimizers still open?

Let $A(k,n)$ be the set of $\{0,1\}$ matrices of order $n$ with all their line sums equal to $k$. Conjecture number 5 on the list from Minc's book, attributed to Ryser, says that if $A(k,n)$ ...
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### On the convexity of element-wise norm 1 of the inverse

Question first asked on math.stackexchange here: http://math.stackexchange.com/questions/317209/on-the-convexity-of-element-wise-norm-1-of-the-inverse On the convexity of element-wise norm 1 of the ...
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### Fastest 'Oracle' Algorithm for satisfying a single linear constraint on a convex set?

In this paper by Arora, Hazan, and Kale (http://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf) a method is given for using the Multiplicative Weights Update algorithm to quickly solve Linear Programs ...
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### How do you prove that every curve of constant width is convex?

Cross-posted from math.stackexchange: Let C be a simple closed plane curve and let D be its interior. Recall that the width of C in a direction θ is the distance between two supporting lines for D ...
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### Support Functions Of 3D Convex Bodies In Spherical Polar Coordinates

What are the known sufficient conditions,analagous to the planar curvature condition, in terms of functions of theta and phi, on the support function h(theta,phi) of a surface in 3D which imply it is ...
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### Covering the cone of positive semidefinite matrices by intervals

Is it possible to cover the cone of positive semidefinite matrices by a finite/countable/interesting family of closed intervals of matrices? How about a general convex cone? For the finite case the ...
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### Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equalities between Distance Sums or Differences

Given 4 points $A$, $B$, $C$ and $D$ in general position in the euclidean plane, is it possible to determine from the 6 distances $AB$, $BC$, $CD$, $AD$, $AC$ and, $BD$ alone, whether every point is a ...
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### another diameter-perimeter-area inequality

Recently I learnt that $$\inf\frac{diam(C)(per(C)-2diam(C))}{area(C)}=0$$ where the infimum is taken over all plane convex bodies $C$ (say, with non-zero area). In other words, there is no non-...
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### Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$?

In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$. Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are ...
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### Duality between extremal points and extremal maps

Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ and every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set ...
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### Is these two optimization problems share the same solution?

Hello all, I am dealing with some SDP optimization, and I come across the following problem. The optimization problem is given by \begin{aligned} &\operatorname*{min}_{t_1,\ldots,t_m,X}\ \sum ...
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### Eigenvalues of product of two symmetric matrices

This is mostly a reference request, as this must be well known! Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$...
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### Question regard checking convexity by “restriction to any line that intersects the function domain”

Hello all, I have a question (probably stupid one) about the fact that " A function is convex if and only if it is convex when restricted to any line that intersects its domain". In Stephen Boyd and ...
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### Is a jointly convex function of x and y convex as a function of x when y=z(x)?

Hi, Suppose that $x \in R^m, y \in R^n, z(x) \in R^n$, and $f(x,y)$ is convex in $(x,y)$. Is $f(x,z(x))$ a convex function in $x$ for arbitrary continuous functions $z(x)$? Thanks!