All Questions
5 questions
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Converse of Scherk–Segre theorem on the number of vertices of a convex space curve
It is well known that any smooth simple closed convex curve $\gamma$ in $\mathbb{R}^{3}$ that meets no plane in more than 4 points has exactly 4 vertices, i.e., points of vanishing torsion; here "...
4
votes
1
answer
139
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Characterization of convexity by connectedness of hyperplane sections
Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$.
Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is ...
12
votes
1
answer
281
views
Rigidity of doubled convex caps
Suppose that we have a convex cap, i.e., a convex surface in $R^3$ homeomorphic to a disk whose boundary lies in a plane. Reflect the cap through the plane of its boundary and glue it back to the ...
2
votes
1
answer
561
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Quermassintegrals as mean curvature integrals
It is well-known that the quermassintegrals $W_{i}$ of a convex body $K\subset \mathbb{R}^{n}$ can be written as a mean curvature integral
$$ W_{i}(K)=n^{-1}\int_{\partial K}H_{i-1} d\mathcal{H}^{n-1},...
8
votes
1
answer
4k
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Geometric meaning of derivatives of the curvature
Let $\Gamma$ be a sufficiently smooth, closed, convex curve in the plane parametrized by arclength $s$. Further assume that $\kappa>0$ and that the second derivative of $\kappa$ with respect to $s$ ...