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OPEN PROBLEMS IN GEOMETRY OF CURVES AND SURFACES
We collect dozens of well-known and not so well-known fundamental unsolved problems involving low dimensional submanifolds of Euclidean space. The list includes selections from differential geometry,Expand
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Circles minimize most knot energies
Abstract We define a new class of knot energies (known as renormalization energies ) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knotExpand
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Optimal Smoothing for Convex Polytopes
It is proved that given a convex polytope P in R, together with a collection of compact convex subsets in the interior of each facet of P , there exists a smooth convex body arbitrarily close to PExpand
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The problem of optimal smoothing for convex functions
A procedure is described for smoothing a convex function which not only preserves its convexity, but also, under suitable conditions, leaves the function unchanged over nearly all the regions whereExpand
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Totally skew embeddings of manifolds
We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an n-dimensional submanifold without any pairs of parallelExpand
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Shortest periodic billiard trajectories in convex bodies
AbstractWe show that the length of any periodic billiard trajectory in any convex body $$ K \subset \mathbf{R}^n $$ is always at least 4 times the inradius of K; the equality holds precisely whenExpand
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h-Principles for curves and knots of constant curvature
We prove that $${\mathcal{C}}^\infty$$ curves of constant curvature satisfy, in the sense of Gromov, the relative $${\mathcal{C}}^1$$-dense h-principle in the space of immersed curves in EuclideanExpand
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Strictly Convex Submanifolds and Hypersurfaces of Positive Curvature
We construct smooth closed hypersurfaces of positive curvature with prescribed submanifolds and tangent planes. Further, we develop some applications to boundary value problems via Monge-Amp ereExpand
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The relative isoperimetric inequality outside convex domains in Rn
We prove that the area of a hypersurface Σ which traps a given volume outside a convex domain C in Euclidean space Rn is bigger than or equal to the area of a hemisphere which traps the same volumeExpand
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The convex hull property and topology of hypersurfaces with nonnegative curvature
Abstract We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed hypersurface lies outside the convex hull of its boundary, provided the boundary satisfies certainExpand
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