Three Elementary Planar Convexity Problems

[IanCalvert]

It is well known that for compact subsets, X, of the Euclidean plane,
with interior points, X is convex if and only if each frontier point
has at least one support line: see Eggleston, [1], Theorems 8 and 9.
For other relevant convexity terminology and background also see [1].

Assume in the rest of this communication that X is a planar, compact
convex set with interior points.

If p is a fixed interior point of X, let f(x, p) be the foot of the
perpendicular from p to any support line through a frontier point x of
X. If there is more than one support line to X at x define f(x, p) as
the foot that maximises the f/g distance ratio below.

Assuming now p and f(x, p) are known, define g(x, p) as the unique
frontier point of X on the line between p and f(x, p). Note that (i)
g(x, p), f(x, p) and x may coincide if X is a closed disc and p is its
centre (ii) if X is a square and p its centre then the choice of
support line at the corners is important.

Now consider the ratio of the distance of f(x, p) from p divided by
the distance of g(x, p) from p. This f/g distance ratio is greater
than or equal to 1.

Not only is this ratio 1 for closed discs if p is the centre, in fact
this property characterises the centre of balls in normed spaces.

Zeev Smilansky published a simple proof (of the planar case) of that
characterisation in [2]. He also gave various characterisations of
the centre of a ball in a sadly unpublished paper included with a
private communication March 1986. This followed my 1984 presentation,
of a different proof, to a splinter group at the British Mathematical
Colloquium (BMC) in Bristol.

As p approaches the frontier of X the f/g distance ratio goes to
infinity so here it is appropriate to impose a condition that p is
�well inside� X. One natural, but of course by no means the only
possible, choice is to let p be the circumcentre of X which is assumed
from now on.

The three problems of the communication title then involve considering
three types of compact convex X:

(i) unrestricted convex X
(ii) central sets X
(iii) sets of constant width X

For each type, what is the maximum value of the f/g distance ratio for
fixed p over varying x and what are the extremal convex sets?

I believe the answers are:
(i) Equilateral Triangles with a value 1.500
(ii) Squares with a value 1.207
(iii) Reuleaux Triangles with a value 1.116

However I am unable to prove or refute these conjectures.

Comments and references to any relevant literature would be welcome.

References

1. H.G.Eggleston, Convexity, Cambridge University Press, 1969

2. Problem 71.D, Problem Corner, Mathematical Gazette, December 1987

[IanCalvert]

On 1 Mar, 22:30, IanCalvert <ical12...@btinternet.com> wrote:
> It is well known that for compact subsets, X, of the Euclidean plane,
> with interior points, X is convex if and only if each frontier point
> has at least one support line: see Eggleston, [1], Theorems 8 and 9.
> For other relevant convexity terminology and background also see [1].
>
> Assume in the rest of this communication that X is a planar, compact
> convex set with interior points.
>
> If p is a fixed interior point of X, let f(x, p) be the foot of the
> perpendicular from p to any support line through a frontier point x of
> X. If there is more than one support line to X at x define f(x, p) as
> the foot that maximises the f/g distance ratio below.
>
> Assuming now p and f(x, p) are known, define g(x, p) as the unique

> frontier point of X on the line between p and f(x, p). �Note that (i)

> g(x, p), f(x, p) and x may coincide if X is a closed disc and p is its
> centre (ii) if X is a square and p its centre then the choice of
> support line at the corners is important.
>
> Now consider the ratio of the distance of f(x, p) from p divided by
> the distance of g(x, p) from p. This f/g distance ratio is greater
> than or equal to 1.
>
> Not only is this ratio 1 for closed discs if p is the centre, in fact
> this property characterises the centre of balls in normed spaces.
>
> Zeev Smilansky published a simple proof (of the planar case) of that

> characterisation in [2]. �He also gave various characterisations of

> the centre of a ball in a sadly unpublished paper included with a
> private communication March 1986. This followed my 1984 presentation,
> of a different proof, to a splinter group at the British Mathematical
> Colloquium (BMC) in Bristol.
>
> As p approaches the frontier of X the f/g distance ratio goes to
> infinity so here it is appropriate to impose a condition that p is

> �well inside� X. One natural, but of course by no means the only

> possible, choice is to let p be the circumcentre of X which is assumed
> from now on.
>
> The three problems of the communication title then involve considering

> three types of �compact convex X:
>
> (i) � � unrestricted �convex X
> (ii) � �central sets �X
> (iii) � sets of constant width X

>
> For each type, what is the maximum value of the f/g distance ratio for
> fixed p over varying x and what are the extremal convex sets?
>
> I believe the answers are:

> (i) � � � Equilateral Triangles with a value 1.500
> (ii) � � �Squares with a value 1.207
> (iii) � � Reuleaux Triangles with a value 1.116

>
> However I am unable to prove or refute these conjectures.
>

> Comments �and references to any relevant literature would be welcome.
>
> References
>
> 1. � H.G.Eggleston, Convexity, Cambridge University Press, 1969
>
> 2. �Problem 71.D, Problem Corner, Mathematical Gazette, December 1987

D'oh! and sincere regrets at errors due to haste in posting.

Long and thin rectangles show that both (i) and (ii) are incorrect.

Although it is possible to avoid that case by restricting the minimal
width of X it is probably best to consider case (iii) only.