Convexity of Suns in Tangent Directions

A direction

is called a tangent direction to the unit sphere

S S

if the conditions

s\in S s ∈ S

and

\operatorname{aff}(s+d) aff ⁡ ( s + d )

is a tangent line to the sphere

S S

at

s s

imply that

\operatorname{aff}(s+d) aff ⁡ ( s + d )

is a one-sided tangent to the sphere

S S

, i.e., it is the limit of secant lines at the point

s s

. A set

M M

is called convex with respect to a direction

d d

if

[x,y]\subset M [ x , y ] ⊂ M

whenever

x,y\in M x , y ∈ M

,

(y-x)\parallel d ( y − x ) ∥ d

. It is shown that in an arbitrary normed space an arbitrary sun (in particular, a boundedly compact Chebyshev set) is convex with respect to any tangent direction of the unit sphere.