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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 "...
Matteo Raffaelli's user avatar
4 votes
1 answer
139 views

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 ...
Dmitrii Korshunov's user avatar
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 ...
Mohammad Ghomi's user avatar
2 votes
1 answer
561 views

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},...
Sebastian Scholtes's user avatar
8 votes
1 answer
4k views

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 $s$ ...
user17240's user avatar
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