# Questions tagged [convex-geometry]

A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...

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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 ...
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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 ...
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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 ...
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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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### Universal covering problem in dimension three

In dimension two one can quickly prove that every shape of diameter $1$ can be covered with a regular hexagon of width $1$. Thus, one may ask what is the minimal area of such a "universal ...
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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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### 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 ...
1 vote
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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 ...
1 vote
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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 vote
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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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### 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 ...
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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 ...
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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 ...
1 vote
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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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### 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. $p_i$ ...
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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 ...