Upper semicontinuity of the lamination hull
| Autor: | Terence L. J. Harris |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Convex hull
0209 industrial biotechnology Control and Optimization 010102 general mathematics Dynamical Systems (math.DS) 02 engineering and technology Lamination (topology) 01 natural sciences Set (abstract data type) Combinatorics Computational Mathematics Mathematics - Analysis of PDEs 020901 industrial engineering & automation Compact space Control and Systems Engineering Hull Diagonal matrix FOS: Mathematics Symmetric matrix Mathematics - Dynamical Systems 0101 mathematics Analysis of PDEs (math.AP) 49J45 (Primary) 52A30 (Secondary) Mathematics |
| Zdroj: | ESAIM: Control, Optimisation and Calculus of Variations. 24:1503-1510 |
| ISSN: | 1262-3377 1292-8119 |
| DOI: | 10.1051/cocv/2017033 |
| Popis: | Let $K \subseteq \mathbb{R}^{2 \times 2}$ be a compact set, let $K^{rc}$ be its rank-one convex hull, and let $L(K)$ be its lamination convex hull. It is shown that the mapping $K \to \overline{L(K)}$ is not upper semicontinuous on the diagonal matrices in $\mathbb{R}^{2 \times 2}$, which was a problem left by Kol\'a\v{r}. This is followed by an example of a 5-point set of $2 \times 2$ symmetric matrices with non-compact lamination hull. Finally, an example of another 5-point set $K$ is given, which has $L(K)$ connected, compact and strictly smaller than $K^{rc}$. Comment: 8 pages, 2 figures. Accepted version |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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