All Questions
1,544 questions
1
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Membership test of convex set
Let $K$ be a compact convex subset of $R^n$ which has some positive gaussian measure, say at least 1/2. For each nonzero vector $u \in R^n$, we
define another compact convex set $K * u$ in the ...
- 11
4
votes
2
answers
902
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Does every Banach space admit a continuous (not necessarily equivalent) strictly convex norm?
Trying to find and answer to this question, I have encountered two more-studied problems.
The first is to find when a Banach space admits an equivalent uniformly convex norm. The answer is that for ...
- 1,955
0
votes
1
answer
89
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Iterated barycentric subdivision cofinal in system of subdivisions?
Given a rational polyhedral fan $\Sigma$ in $\mathbb{R}^d$ (say with full-dimensional support), its barycentric subdivision $\mathrm{bar}(\Sigma)$ is obtained by performing star-subdivision at the ...
- 691
3
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0
answers
76
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A claim on planar sections of 3D convex bodies
Ref: More on shadows of 3D convex bodies,
Shadows and planar sections of polyhedra
Given a 3D convex body C, we define a maximal area (perimeter) section of C with respect to any specified direction $...
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2
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0
answers
114
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More on shadows of 3D convex bodies
Ref: Shadows and planar sections of polyhedra
By shadow we mean the orthogonal projection of a convex 3D body C onto a 2D plane, for example, the shadow on the xy-plane, with C above (z>0) that ...
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2
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1
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78
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On convex solids with all plane sections affine congruent
Question: How many (classes of) convex 3D solids are there such that all non-degenerate planar sections of the solid are mutually affine congruent?
Further question: Same as above with 'projective' ...
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4
votes
2
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217
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On faces of polytopes
$\newcommand\ext{\operatorname{ext}}\newcommand\R{\mathbb R}$Let $A$ be a convex polytope in $\R^n$ with nonempty interior.
Consider the closed convex cone
$$K_A:=\{(l,t)\in(\R^n)'\times\R\colon\, l(x)...
- 128k
6
votes
3
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551
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Hahn-Banach Theorem for convex polytopes and their supporting hyperplanes
A polytope in $\mathbb R^n$ is the convex hull of a nonempty finite set in $\mathbb R^n$.
Let $C$ be a polytope in $\mathbb R^n$.
We shall say that a hyperplane $H\subseteq \mathbb R^n$
$\bullet$ ...
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2
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0
answers
163
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Log Sobolev inequality for log concave perturbations of uniform measure
Suppose $\Omega$ is a convex bounded open set of $\mathbb{R}^n$ (I would be happy with just $\Omega$ as the $n$-dimensional cube). Let $\mu$ be the uniform measure on $\Omega$ and consider the ...
- 873
1
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2
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112
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A claim on concurrency of 'Width Bisectors' of planar convex regions
We add a bit to A claim on the concurrency of area bisectors of planar convex regions
Define a width of a planar convex region $C$ as the distance between two parallel lines that just touch $C$. A ...
- 5,979
1
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0
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61
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$\psi_2$ marginals of the permutahedron?
Let $K$ be a convex body.
I in particular care about the permutahedron.
I will view this as being the convex hull of all coordinate-wise permutations of the vector
$$v = \frac{1}{2n+2}(-n, -n+2,\dots, ...
- 2,183
1
vote
1
answer
147
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Link between asymptotic cone and the boundary of a convex set
For $n\geq 1$, let $f\in\mathcal{C}^2(\mathbb{R}^n ; \mathbb{R})$ such that $f^{-1}(\mathbb{R}_-^*)\neq\varnothing$ and $\forall x\in\mathbb{R}^n$: $\nabla f(x)\neq 0$ and $\mathcal{Hess}f(x)$ is ...
- 449
1
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52
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On families of lines that cut the boundary of a planar convex region in a specified ratio
We proceed from A claim on the concurrency of area bisectors of planar convex regions
This question is somewhat broad.
Background: 'Mathematical Omnibus' by Fuchs and Tabachnikov, Lecture 11 describes ...
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10
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2
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489
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Does approximate equality of quantum states imply operator inequality in a large subspace?
Let the trace norm of $X$ be
$$\Vert X\Vert_1 := \operatorname{tr} \left(\,(X^\dagger X)^{1/2}\right)$$
and let the operator inequality $A \leq B$ denote that the operator $B-A$ is positive ...
- 165
1
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2
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157
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A claim on the concurrency of area bisectors of planar convex regions
We add a little bit to On 'fair bisectors' of planar convex regions and Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia
Definitions: Given a ...
- 5,979
5
votes
0
answers
158
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Log Sobolev inequality uniform in parameters
Fix a positive integer $N$. For $\theta \in [0,2\pi]$, set $\sigma_k(\theta) :=(\cos(k\theta),\sin(k\theta)) \in S^1$ for each integer $1\leq k\leq N$. Now for vectors $x_1,\ldots,x_N\in \mathbb{R}^2$,...
- 873
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2
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530
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Any idea of solving an optimization problem with cubic constraints?
I have the following optimization problem with cubic constraints, which is hard to solve. Are there any ideas, or related references, of solving such a problem?
$$ \begin{array}{ll} \underset {y, z} {\...
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18
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3
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997
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Convex functions in convex sets
Suppose $\Omega \subset \mathbb{R}^n$ is some bounded, convex set. For which domains $\Omega$ is it true that for every convex function $f:\Omega \rightarrow \mathbb{R}$ the average of the function in ...
2
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0
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66
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Understanding a claim of Makai and Martini: why is an ellipsoid's cross-section body the same as its projection body?
In The Cross-Section Body, Plane Sections of Convex
Bodies and Approximation of Convex Bodies, I (Makai and Martini, 1996), the authors define for a convex body $K$ its cross-section body $CK$ and its ...
- 3,068
4
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1
answer
191
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Sliding a convex body over a Gaussian measure
Consider an $n$-dimensional convex set $K \subset \mathbb{R}^n$ and let $\mu$ denote the Gaussian measure with density
$$
\gamma(\mathbf{x}) = \frac{1}{(2\pi)^{n/2}} e^{-\lVert \mathbf{x} \rVert^2/2}.
...
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1
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0
answers
83
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Closed form volumes for intersecting modified cylinders
This question is somewhat related to the question Intersecting cylinders, but where the cylinders are now modified to orbifolds in the hypercube with singularities occurring at the vertices of the ...
- 383
3
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1
answer
107
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A question on strongly convex rational polyhedral cone
Let A be a strongly convex rational polyhedral cone in R^n.
Does (A+(-A))⋂Z^n=(A⋂Z^n)+(-A⋂Z^n) hold?
If it holds, why does it hold?.
- 197
1
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0
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102
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Extreme points of a two-dimensional convex body in terms of its surface area measure
Let $K \subset \mathbb{R}^2$ be a nonempty compact convex set.
For any $t \in S^1$, define the unit vector $u_t = (\cos t, \sin t)$ making an angle of $t$, and let $l_K(t)$ be the tangent line of $K$ ...
- 143
6
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2
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539
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Conditions for including cones
Consider $N$ $n$-dimensional vectors, where the angle between any two vectors is acute and their starting point is at the origin. Can we rotate these vectors together so that the coordinate components ...
- 61
2
votes
1
answer
594
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Tangent cone of a closed convex cone
Let $K \subset \mathbb{R}^n$ be a closed convex set. Given a point $u \in K$, the tangent cone of $K$ at $u$ is defined as (or characterized by)
$$
T_K(u) := \mathrm{cl}(\left\{ t (v - u) \mid v \in K,...
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199
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good textbooks which deal with convex geometry, especially convex polyhedral cones and convex polytopes
I study toric varieties. In toric geometry, we use convex geometry, especially convex polyhedral cones and convex polytopes. Are there good textbooks which deal with convex geometry, especially convex ...
- 197
4
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2
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353
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An upper bound of gradient norm for convex functions near minimizer
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a convex function. Denote $X^*$ as the set of minimizers of $f$ and assume $X^*$ is unbounded. Is it possible that $\|g_x\|$ is unbounded when $d(x,X^*)$ ...
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5
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1
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223
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Is the interior of the tensor product of two convex cones equal to the tensor product of their respective interiors?
I am sorry that the following question is elementary. I have not received an answer from my post at Math Stack Exchange.
In the following question, all cones are convex and contain the origin. Let $C \...
- 331
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0
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81
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Expectation of dual norm induced by probability measure
Let $\mu$ be a probability measure supported on the unit sphere $\mathbb{S}^{d-1}$. Assuming that $\mu$ is even and not supported on any great subsphere, its cosine transform $w \in \mathbb{R}^d \...
- 103
1
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2
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121
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How to solve the optimization problem $\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$?
I am looking for an algorithm to solve the following optimization problem
$$\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$$
where $\mathbf{w}$ and each $\mathbf{x}_i\in\mathbb{R}^d$.
...
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4
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0
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52
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Quantifying error in the reconstruction of convex polytopes from moments
The problem of reconstructing a geometric object from its moments is of interest in a variety of fields. In the paper The Inverse Moment Problem for Convex Polytopes, the authors show that a convex ...
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4
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1
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159
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Approximation of convex bodies by polytopes corresponding to smooth toric varieties
Let $P\subset \mathbb{R}^n$ be an $n$-dimensional polytope with rational vertices. There is a well known construction which produces an $n$-dimensional algebraic variety $X_P$ called toric variety. In ...
- 21.8k
1
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51
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Partitioning convex regions, maximizing the average perimeter of pieces
We continue from Cutting convex regions into equal diameter and equal least width pieces - 2
Question: If a planar convex region C is to be cut into n convex pieces such that the average of the ...
- 5,979
3
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0
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105
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Techniques for solving linear inequalities
For $n$ real variables $x_1, \ldots, x_n$, I have a bunch of inequalities of form $2 x_i > x_j + x_k$ or $2 x_i < x_j + x_k$, where $i,j,k$ are distinct. My goal is to determine whether this set ...
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61
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Fitting a convex polytope with 𝑛 facets between two nested spheres
This is related to a research problem that is interested in approximation of spheres by convex polytopes.
Let $C_r$ and $C_R$ be two spheres in $\mathbb R^d$ of radius $r$ and $R$, respectively, where ...
1
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0
answers
82
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Inside-out dissections of polygons - a generalization
Definitions (Inside-out polygonal dissections): a polygon P has an inside out dissection into P' if P′ is congruent to P and the perimeter of P becomes interior to P' and so the perimeter of P' is ...
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1
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103
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Constrained linear optimization problem on $C^1$
I am dealing with a problem of the form ($a<b$)
$$
\displaystyle \max_{v \in C^1([a, b])} \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \...
5
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1
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227
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The bounded complex of a polyhedral decomposition
Let $\mathscr{P}$ be a polyhedral decomposition of a real vector space $V$. By that I mean that $\mathscr{P}$ is a finite set of polyhedra in $V$ satisfying the following three properties:
The union ...
- 2,537
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0
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47
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On convexity of special fractals in the plane
Let $X \subset \mathbb{R}^2$ be a subset of the plane with Hausdorff dimension $1<dim_H(X)<2$.
For a subset $Y \subset \mathbb{R}^n$ we define $Y$ to be convex if for every $y_1,y_2 \in Y$ the ...
- 1,343
7
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1
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568
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Asking for an English version of Aleksandrov's famous 1939 paper in Convex Geometry
I have difficulty even in finding a Russian version of the next paper:
"Aleksandrov, A. D., Almost everywhere existence of the second differential of a convex
function and some properties of ...
- 300
1
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0
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36
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Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?
Suppose that $P$ is a polyhedron represented by
$$P:=\{x \in \mathbb{R}^n: A x \le b \} \text{ for }A \in \mathbb{R}^{m\times n},\ b \in \mathbb{R}^m,$$
and $P$ contains interior points. Moreover, the ...
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0
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145
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Bound on solutions of $Ax \ge b$
Let $A \in \mathbb{Z}^{m \times n}, b \in \mathbb{Z}^{m \times 1}$.
One can show that if there is a solution of $Ax \ge b, x \in \mathbb{R}^n$ then there is one such that $\|x\|_{\infty} \le c (\|A\|_{...
- 439
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1
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101
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Maximum number of vectors with bounds on inner products (follow up question)
This is a follow-up question from my previous question.
Suppose there are (2n+1) vectors $\{m_1,m_2,...,m_n\}$, $\{p_1,p_2,...,p_n\}$ and $p^*$ in $R^{k+1}$. $m_i$ are weakly positive vectors. $p_i$ ...
- 23
0
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0
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84
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1-degree SOS proof refutes Linear Programming
I am trying to understand Sums-of-Squares proof systems.
A degree $d$ Sums-of-Squares refutation for a set of polynomial equations $P = \{p_1(x) = 0, ..., p_m(x) = 0\}$ is defined as
$\sum_{i=1}^m g_i(...
1
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1
answer
119
views
Adding linear constraint to the domain
I don't know if it is a well-known problem, but I have been struggling to come up with an algorithm.
I have a set of linear constraints $Ax\le b$, $b\ge 0$ ($b$ and $A$ are given, $x$ is a variable). ...
1
vote
1
answer
176
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Is an inner product $\langle X, \epsilon\rangle$ between log-concave $X$ and $\epsilon\gets \{0,1\}^n$ log concave?
Let $X$ be a random variable with a density $p(x)$ with respect to the Lebesgue measure. We say that $X$ is log concave if $p(x) = \exp(-V(x))dx$ for $V(x)$ a convex function.
Let $X$ be log-concave ...
- 2,183
0
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0
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140
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Support function of the intersection of a hyper-ellipsoid and a Euclidean ball
Le $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_d$ be positive numbers. For any $x \in \mathbb R^d$ and $r \ge 0$, define $\gamma(x,r) := \sup_{z \in E(r)}x^\top z$, where
$$
E(r) := E \cap B_2^d(r)...
- 6,853
0
votes
0
answers
74
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The closest ellipse and circle to a given triangle - 2
We add a little more to The closest ellipse to a given triangle.
The above linked discussion used the Hausdorff distance to quantify how close two planar convex regions are.
In an earlier post - ...
- 5,979
8
votes
2
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287
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Does the surface area of the unit Lp ball go to zero for all $p < \infty$?
We know about volume: The $L_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of ...
- 4,529
2
votes
0
answers
200
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Toric decomposition of multipartitions
Fix $k \in \mathbb Z_{>0}$. By a $k$-multipartition $\lambda=(\lambda_1,\dots,\lambda_k)$ of $N$, I mean that each $\lambda_i$ is a partition of some $N_i$ and $\sum N_i = N$.
Let's call $\lambda$ ...
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