The Space of Convex Bodies

Gruber, Peter M.

doi:10.1016/b978-0-444-89596-7.50014-6

Cited by 53 publications

(16 citation statements)

References 79 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The map t → U t is continuous when the set of convex domains is endowed with the Hausdorff convergence (see Henrot & Pierre (2005) or Gruber (1993) for more details on this convergence). By continuity of the volume and the Dirichlet eigenvalues for the Hausdorff convergence of convex sets (see theorem 2.3.17 in Henrot 2006 for the eigenvalues), the continuous path…”

Section: Proc R Soc a (2011)mentioning

confidence: 99%

On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

Antunes

Henrot

2010

Proc. R. Soc. A.

View full text Add to dashboard Cite

In this paper, we study the set of points in the plane defined by {(x, y) = (l 1 (U), l 2 (U)), |U| = 1}, where (l 1 (U), l 2 (U)) are either the first two eigenvalues of the Dirichlet-Laplacian, or the first two non-trivial eigenvalues of the Neumann-Laplacian. We consider the case of general open sets together with the case of convex open domains. We give some qualitative properties of these sets, show some pictures obtained through numerical computations and state several open problems.

show abstract

Section: Proc R Soc a (2011)mentioning

confidence: 99%

On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

Antunes

Henrot

2010

Proc. R. Soc. A.

View full text Add to dashboard Cite

show abstract

“…The GL(n)-space B(n) is of a special interest in Geometric theory of finite-dimensional Banach spaces [19, §5.2], as well as in Convex geometry [15]. The orbit space B(n)/GL(n) is just the famous Banach-Mazur compactum BM(n) and it is an interesting topological object (see [19, p. 1191] and [25, Problem 899]).…”

Section: Corollary 46 Let G Be a Lie Group And X Be A Convex Invarimentioning

confidence: 99%

Extending equivariant maps into spaces with convex structure

Antonyan¹

2005

Topology and its Applications

View full text Add to dashboard Cite

We prove that if G is a compact Lie group, Y a G-space equipped with a topological local convex structure compatible with the action of G, then Y is a G-ANE for metrizable G-spaces. If, in addition, Y has a G-fixed point and admits a global convex structure compatible with the action of G, then Y is a G-AE. This is applied to show that certain hyperspaces related to the Banach-Mazur compacta are equivariant absolute extensors.

show abstract

“…Now we mention some properties of the functionals H and D. The proofs follow directly and straightforwardly from the associated properties of the support functions γ S . The functional H introduces the Hausdorff distance and both H and the underlying support functions σ S are widely studied [5], [10], [13], [17], which is not the case for D and ι S respectively. We see by definition that both H(q 1 , S) and D(q 1 , S) reduce to the distance q 1 − q 2 when S is the singleton {q 2 }.…”

Section: Support and Distance Functionalsmentioning

confidence: 99%

“…While the Hausdorff distance H (q, S) appears often in the literature, e.g. [14] or [10], concerning D(q, S) mostly the case D(q, S) ≥ 0 is studied in relation to the metric projection. Some prerequisites for the case of an arbitrary D(q, S) are given in [12] and the one-dimensional case Y = R appeared in [9].…”

Section: Introductionmentioning

confidence: 99%

Approximation of set-valued functions by single-valued one

Ginchev¹,

Hoffmann²

2002

Discuss. Math. Differential Inclusions, Control and Optimizatio

View full text Add to dashboard Cite

Let Σ : M → 2 Y \ {∅} be a set-valued function defined on a Hausdorff compact topological space M and taking values in the normed space (Y, •). We deal with the problem of finding the best Chebyshev type approximation of the set-valued function Σ by a singlevalued function g from a given closed convex set V ⊂ C(M, Y). In an abstract setting this problem is posed as the extremal problem sup t∈M ρ(g(t), Σ(t)) → inf, g ∈ V. Here ρ is a functional whose values ρ(q, S) can be interpreted as some distance from the point q to the set S ⊂ Y. In the paper, we are confined to two natural distance functionals ρ = H and ρ = D. H(q, S) is the Hausdorff distance (the excess) from the point q to the set cl S, and D(q, S) is referred to as the oriented distance from the point q to set cl conv S. We prove that both these problems are convex optimization problems. While distinguishing between the so called regular and irregular case problems, in particular the case V = C(M, Y) is studied to show that the solutions in the irregular case are obtained as continuous selections of certain set-valued maps. In the general case, optimality conditions in terms of directional derivatives are obtained of both primal and dual type.

show abstract

The Space of Convex Bodies

Cited by 53 publications

References 79 publications

On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

Extending equivariant maps into spaces with convex structure

Approximation of set-valued functions by single-valued one

Contact Info

Product

Resources

About