A Generalization of a Classical Geometric Extremum Problem

Let

be the boundary of a compact convex body

\mathcal{C} C

in

\mathbb{R}^n,\, n\geq 2 R n ,   n ≥ 2

, and

O O

be an interior point of

\mathcal C C

. Every straight line

l l

containing

O O

cuts from

\mathcal{C} C

a segment

[AB] [ A B ]

with end-points on

\partial \,\mathcal{C} ∂   C

. It is shown that if

[AB] [ A B ]

is the shortest such segment, then

\partial \,\mathcal{C} ∂   C

is smooth at the points

A A

and

B B

(i.e. at both of them there is only one supporting hyperplane for

\mathcal{C} C

) and, something more, the normals to the unique supporting hyperplanes at the points

A A

and

B B

intersect at a point belonging to the hyperplane through

O O

which is orthogonal to

[AB] [ A B ]

. If

\mathcal{C} C

is a smooth compact convex body in

\mathbb{R}^n,\, n\geq 2 R n ,   n ≥ 2

, the above property holds also when

[AB] [ A B ]

is the longest such segment. Similar results are also valid when

O O

is outside the set

\mathcal{C} C

. The “local versions” of these results (when the length

|AB| ∣ A B ∣

of the segment

[AB] [ A B ]

is locally maximal or locally minimal) are valid as well. More specific results are obtained in the particular case when

\mathcal{C} C

is a convex polytope