# Questions tagged [convex-geometry]

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### Convex Body with minimum expected l2 norm

Consider a convex body $K$ centered at origin and symmetric (i.e. if $x\in K$ then $-x\in K$). I desire to find a different convex body $L$ such that $K\subseteq L$ and the following measure is ...
• 1,584
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### Computing the Löwner-John ellipsoid of a polyhedron

The Löwner-John ellipsoid of a convex set $C$ is the minimum-volume ellipsoid (MVE) that encloses it. The ellipsoid can be computed using Khachiyan's method, and there are a number of approximations ...
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### Learning triangles in the plane

I assigned my students the problem of finding a triangle consistent with a collection of $m$ points in $\mathbb{R}^2$, labeled with $\pm1$. (A triangle $T$ is consistent with the labeled sample if $T$ ...
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• 1,429
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### Which convex polytopes have volumes of polynomial bit-length?

A convex polytope is described as an intersection of halfspaces, given as inequalities between linear combinations of variables with rational coefficients. The volume computation problem for convex ...
• 9,232
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• 1,591
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### Restriction of a convex function to {0, 1}^n

Suppose I have a real-valued convex function $f$ on the unit hypercube $[0,1]^n$, and let $\bar{f}$ be its restriction to the integer points $\{0,1\}^n$. Does $\bar{f}$ satisfy any properties, or can ...
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133 views

### Balanced partitioning of a set of axis-parallel 2D rectangles

Fix a constant $0<\alpha<1/2$. The problem is the following. Suppose there are $N$ axis-parallel rectangles on the 2D plane with weights $w_1, w_2,\ldots, w_N$ and with coordinates all in the ...
• 41
146 views

### Batch membership testing for convex polyhedron specified in vertex representation

I have a convex shape defined by a set of vertices (the so-called vertex representation of a convex polyhedron). I also have a large set of points and I would like to test which are contained in the ...
• 3,902
109 views

### Strongly polynomial time algorithm for shortest convex combination

Problem: Let $S$ be a finite set of vectors. Let $C$ be their convex hull. Compute $\operatorname{argmin}_{x \in C} \|x\|$. Reference 1 gives an algorithm for this problem that is finite-time (Section ...
• 662
238 views

### Minimal number of hyperplanes needed to separate sets of points from one other set

Let $\mathbb{R}^d$ be our space. We have a single good set of points $g$, and a collection of bad sets of points $B$. We assume that for all $b \in B$ the convex hulls of $g$ and $b$ are disjoint. ...
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### Efficient algorithm for computing equally distributed points in polytope?

Given a $d$-dimensional (rational) convex polytope $P$ with vertices $v_1, \ldots, v_n$, I would like to compute many equally distributed points that lie in $P$. "Equally distributed" is of course not ...
• 541
277 views

### Algorithm for testing if a point belongs to a sequence of convex set or not

I have a sequence of convex sets $C_\lambda$ for $\lambda \in \mathbb{R}$ such that for $\lambda_1 > \lambda_2$, we have $C_{\lambda_1} \subset C_{\lambda_2}$ (essentially a nested sequence of ...
• 151
83 views

### Enforcing general position in $2d$ linear programming

Let $(x_1, y_1), ..., (x_k, y_k)$ be $n$ points in $\Re^2$. For my sake, $k=20$. I am trying to set up a linear program to find a set of $k$ points in the plane $P$ that satisfy some linear ...
• 1,143
124 views

### Equivalence of weighted Minkowski sums

Given $n$ polytopes $P_1, \cdots, P_n$, each $P_i$ is given as the V-representation, i.e., a set of $m$ points as its set of vertices. Furthermore, consider a variant of the Minkowski sum (somehow ...
189 views

### Computing the convex hull of several polyhedra

Let $K_1,..., K_m$ be a set of $m$ polyhedra, with $K_i\subseteq \mathbf{R}^n$, for all $i\in [m]$, and each is described by a set of $poly(n)$ linear inequalities. How easy is it to compute an ...
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263 views

### When can a convex function induce submodularity?

Say I have a real valued convex function $f$ on the hypercube $[-1,1]^n$. Let $f'$ be the induced function on the discrete hypercube $\{-1,1\}^n$. Now I want to find a vertex on $\{-1,1\}^n$ on which ...
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### Finding a cell in an arrangement of simplices

My question is n-dimensional, but I will begin by dropping the problem down to two dimensions for clarity's sake. It regards defining what is a solution by defining one or more data points that are ...
• 123
149 views

### Is the following problem in $coNP$?

Given an $n\times n$ matrix $M$ with $\mathbb Z$ entries is 'does an $\frac n2\times\frac n2$ minor of $M$ vanish?' in $\bf{coNP}$? At least one $\frac n2\times\frac n2$ minor non-vanish implies rank ...
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• 549
1 vote
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### Detailed exposition for proof of Localization Lemma in paper "Random Walks in a Convex Body and an Improved Volume Algorithm"

I've begun reading the paper "Random Walks in a Convex Body and an Improved Volume Algorithm" by Lovász-Simonovits ('93). Although the paper for the most part is pretty self-contained and ...
1 vote
69 views

### How can we prove what the shortest line between two points avoiding convex obstacles is? (visibility graphs)?

I came across the observation in russell & norvig's artificial intelligence book that the shortest path between two points while avoiding convex polygonal obstacles is a sequence of line segments ...
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1 vote
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### doubt about volume packing lemma for intersection of convex sets and lattices (repost from math SE)

Lemma 3.24 of Additive Combinatorics by Tao and Vu states the following: Let $\Gamma \subset \mathbb{R}^d$ be a lattice of full rank, let $V$ be a bounded open subset of $\mathbb{R}^d$, and let $P$ ...
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An extended formulation (EF) of a polytope $P\subseteq \mathbb{R}^d$ is a system of linear constraints $Ex + Fy = g, y\geq 0$ in variables $(x,y)\in \mathbb{R}^{d+r}$ where $E,F$ are real matrices ...