# 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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### Busemann-Petty type problems on complex vector spaces [closed]

We recently published an article on Busemann-Petty type problems (see https://arxiv.org/abs/2404.05630). As we experienced several times that as soon as an article is published, no updates/corrections/...
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### Does convexity of boundary implies geodesic convexity?

I came across the following result (mentioned on Pg. 3 of this talk) that states that If $D$ is an open connected subset of a complete Riemannian manifold with smooth metric then $\partial D$ convex ...
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### Prékopa-Leindler style inequality?

Does anyone know a simple proof of the following Prékopa-Leindler style inequality: If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$,...
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### '$\alpha$-moments' and '$\alpha$-centers' of planar convex regions

We try to proceed from Least area and least perimeter triangles that contain a convex planar region - how different can they be? The partial answer given to the above question shows a convex ...
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### How many unit cubes are needed to 'hide' a unit cube fully in 3D?

Question: What is the smallest number of nonoverlapping unit cubes that can hide a unit cube C - in the sense that every ray emanating from the boundary of C meets the interior or the boundary of one ...
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### On special points within convex solids with all planar sections passing through them having equal area

Question: If within a convex solid body C there is a special point P such that every planar section of C passing through P has the same area, then, can we assert that C is a sphere and P its center? ...
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1 vote
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### Variants of cutting plane method for convex optimization

The cutting plane approach in convex optimization is a general recipe for minimizing a convex function. The argument relies on the fact that using the gradient vector, we can cut the feasible set into ...
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### Set of enclosed convex polyhedra in a graph

Given a straight-line graph embedded in $\mathbb{R}^3$ with known vertex coordinates and edges and no edge intersections, is it possible to find all the enclosed convex polyhedra inside? If so, is ...
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1 vote
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