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- Author or Editor: Gennadiy Averkov x

## Abstract

A convex body *K* in ℝ^{d} is said to be reduced if the minimum width of each convex body properly contained in *K* is strictly smaller than the minimum width of *K*. We study the question of Lassak on the existence of reduced polytopes of dimension larger than two. We show that a pyramid
of dimension larger than two with equal numbers of facets and vertices is not reduced. This generalizes the main result from
[8].

K. Zindler [47] and P. C. Hammer and T. J. Smith [19] showed the following: Let
*K*
be a convex body in the Euclidean plane such that any two boundary points
*p*
and
*q*
of
*K*
, that divide the circumference of
*K*
into two arcs of equal length, are antipodal. Then
*K*
is centrally symmetric. [19] announced the analogous result for any Minkowski plane

*d*-space ℝ

^{d}, R. Schneider [38] proved that if

*K*⊂ ℝ

^{d}is a convex body, such that each shadow boundary of

*K*with respect to parallel illumination halves the Euclidean surface area of

*K*(for the definition of “halving” see in the paper), then

*K*is centrally symmetric. (This implies the result from [19] for ℝ

^{2}.) We give a common generalization of the results of Schneider [38] and Averkov [4]. Namely, let

*d*-dimensional Minkowski space, and

*K*⊂

*K*is halved by each shadow boundary of

*K*with respect to parallel illumination, then

*K*is centrally symmetric. Actually, we use little from the definition of Minkowskian surface area(s). We may measure “surface area” via any even Borel function ϕ:

*S*

^{d −1}→ ℝ, for a convex body

*K*with Euclidean surface area measure

*dS*

_{K}(

*u*), with ϕ(

*u*) being

*dS*

_{K}(

*u*)-almost everywhere non-0, by the formula

*B*↦ ∫

_{B}ϕ(

*u*)

*dS*

_{K}(

*u*) (supposing that ϕ is integrable with respect to

*dS*

_{K}(

*u*)), for

*B*⊂

*S*

^{d −1}a Borel set, rather than the Euclidean surface area measure

*B*↦ ∫

_{B}

*dS*

_{K}(

*u*). The conclusion remains the same, even if we suppose surface area halving only for parallel illumination from almost all directions. Moreover, replacing the surface are a measure

*dS*

_{K}(

*u*) by the

*k*-th area measure of

*K*(

*k*with 1 ≦

*k*≦

*d*− 2 an integer), the analogous result holds. We follow rather closely the proof for ℝ

^{d}, which is due to Schneider [38].