Geometric-combinatorial characteristics of cones
Abstract
It is shown for a proper closed locally compact subset S of a real normed linear space X that(Error rendering LaTeX formula), where
is the R-kernel of S,
denotes the set of regular points of S and(Error rendering LaTeX formula). Furthermore, it is shown for a closed connected nonconvex subset S of X that(Error rendering LaTeX formula), where D is a relatively open subset of S containing the set
of local nonconvexity points of S. If X is a uniformly convex and uniformly smooth real Banach space, then the first of these formulae is shown to hold with the set
of spherical points of S in place of
, and the second one for a closed connected nonconvex set S. For a connected subset S of a real topological linear space L with nonempty
, the set of strong local nonconvexity points of S, it is shown that(Error rendering LaTeX formula), where
is the quasi-
-kernel of S and(Error rendering LaTeX formula), and that the equality holds provided, in addition, S is open. In conjunction with an infinite-dimensional version of Helly's theorem for flats, these intersection formulae generate Krasnosel‘skii-type characterizations of cones and quasi-cones. All this parallels the
research done recently by the author for starshaped and quasi-starshaped sets.
![ker<sub>R</sub>S](http://siba-ese.unile.it/plugins/generic/latexRender/cache/523f63adb7e0a3bf798263f90b76b8c5.png)
![regS](http://siba-ese.unile.it/plugins/generic/latexRender/cache/22b5ac048c1a6897405f54821db5a345.png)
![IncS](http://siba-ese.unile.it/plugins/generic/latexRender/cache/3500e2530c080744eef2ed889b34ad79.png)
![sphS](http://siba-ese.unile.it/plugins/generic/latexRender/cache/34c68925ac47a4b1ce378789a53ef302.png)
![regS](http://siba-ese.unile.it/plugins/generic/latexRender/cache/22b5ac048c1a6897405f54821db5a345.png)
![slncS](http://siba-ese.unile.it/plugins/generic/latexRender/cache/74b1d969d52163bee5f4316cd9012d0e.png)
![qker_{R<sup>○</sup>}S](http://siba-ese.unile.it/plugins/generic/latexRender/cache/4fa688acf924cfd5f65f90ec952bf4c1.png)
![R<sup>○</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/908279dc59f835533f352691d3760b88.png)
DOI Code:
10.1285/i15900932v16n1p59
Full Text: PDF