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Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$. Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any $0<\varepsilon <\va …

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 …

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 …

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 $M$ be a smooth Riemannian manifold without boundary. Let $X\subset M$ be a closed subset which is a smooth submanifold with boundary, $\dim X=\dim M$. Assume that $X$ is locally convex, i.e. any …

EDIT: Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $f\colon \Omega\to\mathbb{R}$ be a function such that for some $\lambda$ the function $f(x)+\lambda |x|^2$ is convex. Assume that the grap …

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 …

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 $M^n$ be the boundary of a convex compact set in $\mathbb{R}^{n+1}$ with non-empty interior. Question 1. Is $M$ geodesically complete, i.e. is it true that every geodesic (= locally shortest path …

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 …

For a convex compact set $K\subset \mathbb{R}^n$ let us denote by $h_K$ its supporting functional $$h_K(\xi):=\sup_{x\in K}\langle\xi,x\rangle.$$ Thus $h_K\colon \mathbb{R}^n\to \mathbb{R}$ is a conv …

Let $\{K_i\}$ be a sequence of convex compact $n$-dimensional subsets in a Euclidean space $\mathbb{R}^n$. Assume $\{K_i\}$ converges in the Hausdorff metric to a convex compact set $K$ which is also …

Let $K$ be a 3-dimensional convex compact subset either in $\mathbb{R}^3$ or in the 3-dimensional real hyperbolic space $\mathbb{H}^3$. Consider its boundary $\partial K$ equipped with the inner (path …

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 …

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 …