# 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 set with no interior contained in hyperplane?

Let $K$ be a convex set in a normed space $X$. Assume that $int(K)=\emptyset$ (norm topology). Must $K$ be contained in some (affine) hyperplane? It's fairly easy to see that this is true in $ℝ^n$, ...
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### Analytic lower-bound for minimal value of $\|x\|^2$ such that $\|Cx-b\|^2 \le c^2$ (a hyperellipsoid)

Let $C$ be an $n \times p$ matrix and $b$ be a column vector of length $n$, and $c>0$. Let $E := \{x \in \mathbb R^p \mid \|Cx-b\| \le c\}$, a hyperellipsoid in nonstandard position. Question 1. ...
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### Ornstein-Ulhenbeck like semigroup for the orthogonal group with a hypercontractive inequality

I am looking for an Ornstein-Uhlenbeck like semigroup $P_t$ and associated generator $\mathcal{L}$ on $G = \operatorname{SO}(n)$ or $\operatorname{O}(n)$ that has a hypercontractive inequality with a ...
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### A self-isometry of the sphere of a strictly convex Banach space that does not move basic vectors

Problem. Let $n\in\mathbb N$, $X$ be a strictly convex $n$-dimensional real Banach space, $S_X=\{x\in X:\|x\|=1\}$ be the unit sphere of $X$, and $e_1,\dots,e_n\in S_X$ be linearly independent points. ...
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### Dual cone operation is an antitone morphism of lattices

Currently I'm reading a classical text Convex Sets by F. Valentine. While reasing it I'm trying to generalize results as much as I can. The chapter on Dual cones made me thinkin of the fallowing ...
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### A sufficient condition for being the boundary of one's convex hull?

Let $A\subset\mathbb R^n$ be such that: every non-zero linear functional is maximized by a unique point of $A$ every point of $A$ is a point where some linear functional achieves its maximum over $A$...
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### Intersections and curvature in the plane

Let $D$ be a nonempty compact convex plane region whose boundary is a smooth curve whose radius of curvature is at most 1 everywhere. Can the boundary of $D$ intersect a circle of radius 1 in more ...
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### Can every 3-dimensional convex body be trapped by a tetrahedral cage?

Although the title question is fairly unambiguous, I give all relevant definitions: $\bullet$ A subset $C$ of $\mathbb{R}^n$ is an $n$-dimensional convex body if $C$ is convex, compact, and has non-...
We define a affine(concave), upper semi continuous function and bounded function $f:X \to \mathbb{R}$, where $X \subset \mathbb{R}^{k}$ is compact and convex set. Assume that $T$ is an affine and ...
It is known that there are at least three simple, closed geodesics on the surface of any smooth convex body $K$ in $\mathbb{R}^3$, the Lusternik-Schnirelmann Theorem (see links below for references). ...