The dimension of the convex kernel and points of local nonconvexity

Stavrakas, Nick M.

doi:10.1090/S0002-9939-1972-0298549-0

The dimension of the convex kernel and points of local nonconvexity
HTML articles powered by AMS MathViewer

by Nick M. Stavrakas

Proc. Amer. Math. Soc. 34 (1972), 222-224

DOI: https://doi.org/10.1090/S0002-9939-1972-0298549-0

PDF | Request permission

Abstract:

Let S be a compact connected subset of ${R^d}$. A necessary and sufficient condition is given to ensure that the dimension of the convex kernel of S is greater than or equal to k, $0 \leqq k \leqq d$. This condition involves a visibility constraint on the points of local nonconvexity of S. As consequences, we obtain new characterizations of the convex kernel of S and the nth-order convex kernel of S.

References

N. E. Foland and J. M. Marr, Sets with zero-dimensional kernels, Pacific J. Math. 19 (1966), 429–432. MR 205149
John W. Kenelly, W. R. Hare Jr., B. D. Evans, and W. H. Ludescher, Convex components, extreme points, and the convex kernel, Proc. Amer. Math. Soc. 21 (1969), 83–87. MR 238183, DOI 10.1090/S0002-9939-1969-0238183-1
Arthur G. Sparks, Characterizations of the generalized convex kernel, Proc. Amer. Math. Soc. 27 (1971), 563–565. MR 279692, DOI 10.1090/S0002-9939-1971-0279692-8

The dimension of the kernel of a starshaped set

14

F. A. Valentine, Local convexity and $L_{n}$ sets, Proc. Amer. Math. Soc. 16 (1965), 1305–1310. MR 185510, DOI 10.1090/S0002-9939-1965-0185510-6