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The Van der Waerden conjecture is a lower estimate of the permanent of a doubly stochastic matrix. In this article in Wikipedia it is stated that Egorychev's proof uses the Alexandrov-Fenchel inequali …

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 …

I think that the perimeter of convex compact sets on the hyperbolic plane is monotone with respect to inclusion. I am looking for a reference to the above fact.

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 …

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 …

The polytope algebra was defined by P. McMullen in "The polytope algebra" Adv. Math. 78 (1989) as follows. Let us denote by $\Pi'_\mathbb{R}$ the quotient of the free abelian group generated by the sy …

Let $\gamma\colon[a,b] \to \mathbb{R}^n$ be a smooth curve. Let $f_i\to f$ be a sequence of convex functions on $\mathbb{R}^n$ converging uniformly on compact subsets to $f$. Let $\hat\gamma(t):=(\gam …

The classical Busemann-Feller lemma in Euclidean space says the following. Let $K\subset \mathbb{R}^n$ be a closed convex set. Then for any point $x\in \mathbb{R}^n$ there exists unique nearest point …

I have heard (if I am not mistaken) that there exists the following conjecture (or theorem?). Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists anoth …

I have heard that there exists the following conjecture (if I am not mistaken). Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists another unit vector …

Let $A,B$ be two convex compact subsets with non-empty interior in a Euclidean space $\mathbb{R}^n$. Let $f\colon A\to B$ be a bijective isometry between them. Does there exist an isometry $F\colo …

The well known Pogorelov’s rigidity theorem says that if two convex closed surfaces in Euclidean 3-space are isometric to each other being equipped with their intrinsic metrics, then they are congruen …

A well known theorem due to A.D. Alexandrov says that any metric on the 2-sphere $S^2$ with curvature at least -1 (in the sense of Alexandrov) can be isometrically realized either as convex surface in …

There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf Any isometry between two closed smooth convex surfaces (e …

There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf Any isometry between two closed smooth convex surfaces in …