 
Summary: Twopoint 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 twopoint 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 halfspaces delimited by H . The twopoint symmetrization HK of a
subset K of R n with respect to H is dened as follows (see Figure 1)
HK = ((K \ H K) \ H ) [ ((K [ H K) \ H + )
HK
H
K
H K
H +
H
