# 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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### Can a neural network with ReLU activation represents exactly all $B$-bounded and $L$-Lipschitz $K$-max-affine functions?

A max-affine function is defined as the maximum over a set of affine functions, which is always convex. More specifically, we define a $K$-max-affine function $f:\mathbb{R}^d\to\mathbb{R}$ that can be ...
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### Optimal unions of planar convex regions

This post continues Optimal intersections between planar convex regions. Question: Given two planar convex polygonal regions $C_1$ and $C_2$, how does one algorithmically find how to place and orient ...
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### On finding optimal convex planar shapes to cover a given convex planar shape

Covering a specific convex shape S with n copies of another specified convex shape S' (which may be different from S) is well studied - for example, https://erich-friedman.github.io/packing/index.html....
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### On ways to measure the difference between two planar convex regions

This earlier post attempted to quantify the difference between a pair of planar convex regions of equal area and perimeter using Hausdorff distance: On comparing planar convex regions of equal ...
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### Lower-bound for $\mathbb E[e^{-b(v^\top X - c)^2}]$, when $X$ is log-concave in high-dimensions

Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that ...
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### Isometric imbedding of a 2-disk into Euclidean 3-space

Let us call a cap the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this ...
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### Is a cap an Alexandrov space?

Let us call a cap the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this ...