Error bounds for cone inclusion problems in Banach spaces are established under conditions weaker than Robinson’s constraint qualification. The results allow the cone to be more general than the origin and therefore also generalize a classical error bound result concerning equality-constrained sets in optimization. Applications in finite-dimensional differentiable inequalities and in tangent co...

1. The tangent cone. We can estimate the directional derivative and gradient of a smooth function quickly and easily using finite difference formulas. While rather inaccurate, such estimates have some appeal, needing neither calculus rules nor even a closed-form expression for the function. In the variational geometry of sets, the role of derivatives and gradients are played by the cones of “ta...

The purpose of this paper is to provide a modified open mapping theorem, which also applies to maps which are not open, but whose intersection with a convex cone is relatively open. We give a constructive proof, which for the special maps considered here may be seen as an alternative to the indirect proof of Frankowska’s more general open mapping principle. Open mapping theorems are of utmost i...

Introduction: Nowadays radiotherapy plays an important role in cancer treatment. Different radiotherapy techniques improvement emphasizes on using of the precise ، appropriate and useful algorithms. one of these techniques are wedged which is used in radiotherapy to compensate missing tissues and create a uniform dose distribution in tissues. The Siemens Artiste linear accelera...

Given A ∈ Ωn, the n-dimensional spectral unit ball, we show that B is a ”generalized” tangent vector at A to an entire curve in Ωn if and only if B is in the tangent cone CA to the isospectral variety at A. If B 6∈ CA, then the Kobayashi–Royden pseudometric is positive at (A;B). In the case of Ω3, the zero set of this metric is completely described.

Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3, and let Γ be a nonempty closed subset of M . The negative case of the Singular Yamabe Problem concerns the existence and behavior of a complete metric ĝ on M\Γ that has constant negative scalar curvature and is pointwise conformally related to the smooth metric g. Previous results have shown that when Γ is a smooth submanifold of d...

Let M be an n-dimensional riemannian manifold. A natural problem in geometry is that of finding k-dimensional submanifolds N ⊂ M with the property that for any bounded open set U in M , the k-volume of N ∩ U is less than or equal to the volume of any other submanifold in M with boundary equal to ∂(N ∩ U). The submanifolds of M with the property above are called area-minimizing. Notice that when...

We give an explicit combinatorial description of the multiplicity as well as the Hilbert function of the tangent cone at any point on a Schubert variety in the symplectic Grassmannian.