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 Nov 8 comment Complexity of detecting a convex body in $\mathbb{R}^n$? @Joel David Hamkins Thanks for your very detailed answer. Indeed, my problem formulation on $\mathbb{R}^n$ is problematic and imprecise. Thanks for pointing it out. I'm glad to see your first two theorems are established by restricting $\mathbb{R}^n$ to a countable dense set, which is what I meant. Nov 7 revised Complexity of detecting a convex body in $\mathbb{R}^n$? added 18 characters in body Nov 7 comment Complexity of detecting a convex body in $\mathbb{R}^n$? @Robert Israel and @Igor Rivin: Thanks for the remark. I've updated my question accordingly. Nov 7 revised Complexity of detecting a convex body in $\mathbb{R}^n$? added 95 characters in body; edited tags; edited title Nov 6 asked Complexity of detecting a convex body in $\mathbb{R}^n$? Sep 26 awarded Commentator Sep 26 comment Upper and Lower Bounds of the Total Surface Area of Convex Polytopes that Partition a Hypercube Thanks Anton Petrunin and Ilya Bogdanov for the nice answer and explanation. To complete the answer, the maximal area of the intersection of a $D$-cube with a $(D-1)$-dimensional hyper plane is $\sqrt{2}L^{D-1}$, where $L$ is the edge length of hypercube. jstor.org/stable/2046239 Sep 25 asked Upper and Lower Bounds of the Total Surface Area of Convex Polytopes that Partition a Hypercube Sep 15 accepted Angle btween Coordinate Vector and Normal Vector of Facet in a Convex Polytope, Asking for a Counterexample Sep 15 comment Angle btween Coordinate Vector and Normal Vector of Facet in a Convex Polytope, Asking for a Counterexample Thanks Ilya Bogdanov for this nice and clear solution. Sep 14 asked Angle btween Coordinate Vector and Normal Vector of Facet in a Convex Polytope, Asking for a Counterexample Sep 14 awarded Scholar Sep 14 accepted Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects? Sep 14 comment Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects? It takes me some time to understand the result. While searching related literatures, I found this "Integral geometry and geometric probability By Luis Antonio Santaló, pp71" books.google.com/… From my point of view, 'Remark''s answer looks like a generalised version of eq(5.10) in $\mathbb{R}^{n}$ when the breadth $a=0$. Aug 26 comment Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects? I just realised that most published results give $\mathbb{E}|T|$ when K is infinity, whereas I'm mostly interested in a convex ploytope tessellation with small number of cells. So maybe this changes everything, what if I want to find $\mathbb{E}F$ when the number of cells $K$ is small, e.g. $K<50$. In particular, is there a way to write $\mathbb{E}F$ as a function (or approximated function) of $n$, dimension of space and $K$, the number of cells? Aug 26 comment Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects? Thanks for your information. I also found [this document] (algo.inria.fr/csolve/vi.pdf) which summaries useful statistics for Poisson-Voronoi Tessellations on $R^1$, $R^2$ and $R^3$. I haven't found any result about $(n-1)$-dimensional measure of the facets of the tessellation. Nevertheless, there are bunch of software can give asymptotic statistics by simulation, e.g. Qhull Aug 25 comment Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects? Thanks for a very detailed answer. As the problem now becomes finding $E|T|$, I'm searching some literatures about the general result on a convex polytopes tessellation. Thanks for your help again. Furthermore, as you mentioned it in the end, I'd like to know more about restricting $L$ to axis-aligned lines. Aug 24 awarded Supporter Aug 24 revised Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects? deleted 37 characters in body; deleted 3 characters in body; added 3 characters in body Aug 24 revised Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects? deleted 14 characters in body