For $K=K_1\bigsqcup ...\bigsqcup K_s$ and a link map $f:K\to {ℝ}^m$ let $K^{\sim }={\bigsqcup }_{i<j}{K}_i\times K_j$, define a map $f^{\sim }:K^{\sim }\to S^{m-1}$ by $f^{\sim }(x,y)=(fx-fy)/|fx-fy|$ and a (generalized) Massey-Rolfsen invariant $\alpha \left(f\right)\in {\pi }^{m-1}\left(K\right)$ to be the homotopy class of $f^{\sim }$. We prove that for a polyhedron K of dimension ≤ m - 2 under certain (weakened metastable) dimension restrictions, α is an onto or a 1 - 1 map from the set of link maps $f:K\to {ℝ}^m$ up to link concordance to ${\pi }^{m-1}\left(K^{\sim }\right)$. If $K_1,...,K_s$ are closed highly homologically connected manifolds of dimension $p_1,...,p_s$ (in particular, homology spheres), then ${\pi }^{m-1}\left(K^{\sim }\right)\cong {\oplus }_{i<j}{\pi }_{p_i+p_j-m+1}^S$.

Let $D$ be a nonempty compact subset of a Banach space $X$ and denote by $S\left(X\right)$ the family of all nonempty bounded closed convex subsets of $X$. We endow $S\left(X\right)$ with the Hausdorff metric and show that there exists a set $ℱ\subset S\left(X\right)$ such that its complement $S\left(X\right)\setminus ℱ$ is $\sigma$-porous and such that for each $A\in ℱ$ and each $\tilde{x}\in D$, the set of solutions of the best approximation problem $\parallel \tilde{x}-z\parallel \to min$, $z\in A$, is nonempty and compact, and each minimizing sequence has a convergent subsequence.

We consider the Robin eigenvalue problem $\Delta u+\lambda u=0$ in $\Omega$, $\partial u/\partial \nu +\alpha u=0$ on $\partial \Omega$ where $\Omega \subset {ℝ}^n$, $n\ge 2$ is a bounded domain and $\alpha$ is a real parameter. We investigate the behavior of the eigenvalues ${\lambda }_k\left(\alpha \right)$ of this problem as functions of the parameter $\alpha$. We analyze the monotonicity and convexity properties of the eigenvalues and give a variational proof of the formula for the derivative ${\lambda }_1^{\text{'}}\left(\alpha \right)$. Assuming that the boundary $\partial \Omega$ is of class $C^2$ we obtain estimates to the difference ${\lambda }_k^D-{\lambda }_k\left(\alpha \right)$ between the $k$-th eigenvalue of the Laplace operator with...