Let $k$ be a natural number. We consider $k$-times continuously differentiable real-valued functions $f\colon E\to \mathbb{R} $, where $E$ is some interval on the line having positive length. For $0<\alpha <1$ let $I_\alpha (f)$ denote set of values $y\in $ whose preimage $f^{-1}(y)$ has Hausdorff dimension at least $\alpha$. how large can (f)$, as $f$ ranges over all from into $\mathbb{R...

Algorithms for shape offsetting are of great importance in computer-aided design (CAD), computer-aided manufacturing (CAM), the numerical control (NC) of machines, computer graphics and related fields. The need for offset shapes arises in applications such as the numerical control of sawing machines or milling machines in the car industry. The problem of shape offsetting can be formulated as fo...

In the paper, the Fenchel problem of level sets in the smooth case is solved by deducing a new and “nice” geometric necessary and sufficient condition for the existence of a smooth convex function with the level sets of a given smooth pseudoconvex function. The main theorem is based on a general differential geometric tool, the space of paths defined on smooth manifolds. This approach provides ...

reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. ABSTRACT Motion segmentation is an important problem in video processing and compression, and in computer vision. It is usually performed by...

In this paper we prove that the level sets of the first non–constant eigenfunction of the Neumann Laplacian on a convex planar domain have only finitely many connected components. This problem is motivated, in part, by the “hot spots” conjecture of J. Rauch. ∗Supported in part by NSF Grant # 9700585-DMS

In this paper we give a characterization of pointwise and uniform convergence of sequences of homogeneous polynomials on a Banach space by means of the convergence of their level sets. Results are obtained both in the real and the complex cases, as well as some generalizations to the nonhomogeneous case and to holomorphic functions in the complex case. Kuratowski convergence of closed sets is u...