From angles to curvature?

Let me start my blog by revisiting my joint paper with Michel Helfgott, “Angles, Area, and Perimeter Caught in a Cubic” (published a year ago in Forum Geometricorum). In that paper we show that the angles of any triangle of area A and perimeter P are bounded by the vertex angles of the two isosceles triangles of area A and perimeter P. In addition to a wish to release this result from the title’s cubic, that is to provide a purely geometrical proof of it, I would like to suggest possible generalizations in two directions:

(I) Similar results for quadrilaterals, pentagons, etc: are there two special n-gons playing, for each n, the role of the two isosceles triangles (in the bounding now of the angles of all n-gons of area A and perimeter P)?

(II) Moving now to convex planar sets, can one replace angles by curvature, and find two special convex sets providing bounds for it? [Possibly an ill-posed question: one may always flatten the set somewhere in order to get a zero curvature — but, assuming smoothness everywhere, the question on the upper bound (on the curvature of a closed curve of area A and perimeter P) remains valid,]

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s


%d bloggers like this: