2,047 research outputs found
The Second Hull of a Knotted Curve
The convex hull of a set K in space consists of points which are, in a
certain sense, "surrounded" by K. When K is a closed curve, we define its
higher hulls, consisting of points which are "multiply surrounded" by the
curve. Our main theorem shows that if a curve is knotted then it has a nonempty
second hull. This provides a new proof of the Fary/Milnor theorem that every
knotted curve has total curvature at least 4pi.Comment: 7 pages, 6 figures; final version (only minor changes) to appear in
Amer.J.Mat
Local Measure of Convex Surfaces induced by the Wiener Measure of Paths
The Wiener measure induces a measure of closed, convex, (d-1)-dimensional,
Euclidean (hyper-)surfaces that are the convex hulls of closed d-dimensional
Brownian bridges. I present arguments and numerical evidence that this measure,
for odd d, is generated by a local classical action of length dimension two
that depends on geometric invariants of the (d-1)-dimensional surface only.Comment: Talk presented at QFEXT09 in Norman, Oklahoma (6 pages, 2 figs.
- …