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Summary: Two-point symmetrization and convexity
Guillaume AUBRUN and Matthieu FRADELIZI
Abstract
We prove a conjecture of R. Schneider: the spherical caps are the only spheri-
cally convex bodies of the sphere which remain spherically convex after any two-
point symmetrization. More generally, we study the relationships between con-
vexity and two-point symmetrization in the Euclidean space and on the sphere.
Introduction
In the following, n is an integer, n 2, the Euclidean norm on R n is denoted by j:j
and the Euclidean unit sphere by S n 1 . Let H be an aĆne hyperplane, denote by
H the re
ection (the orthogonal symmetry) with respect to H and by H + and H
the two closed half-spaces delimited by H . The two-point symmetrization HK of a
subset K of R n with respect to H is de ned as follows (see Figure 1)
HK = ((K \ H K) \ H ) [ ((K [ H K) \ H + )
HK
H
K
H K
H +
H
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