Consider the space of planar sections of any given convex 3D body.

**Basic Question:** What is the lower bound for the ratio
$$\frac{\text{area of section of greatest perimeter}}
{\text{area of section of greatest area}}\ ?$$

And which convex solid gives it?

It appears that right circular cones can be constructed such that the planar section with highest area and the section with highest perimeter are different, indeed perpendicular to each other. E.g. the ratio seems to be $2/\pi$ if the cone's height equals the diameter of its base.

**Generalization:** Consider the set of quantities: {area, perimeter, diameter, width,...}. Take each pair (x,y) of such quantities. What are bounds on the corresponding ratios for those pairs?

(For this, we can define the diameter of a planar convex region as the greatest distance between any pair of points in the region, and the width of a convex region as the least distance between a pair of parallel lines that just touch it.)

**Further questions:** From among bodies for which two quantities x and y from the above set are maximized on the same planar section, will a third quantity z be maximized on another plane? If so how to quantify this difference?

E.g.: From among 3D solids with both area and perimeter maximized on the same planar section, are there any solids whose width is maximized on another planar section? And if so, which solid gives the lower bound on the ratio between the width of the section maximizing both area and perimeter and the width of the section maximizing width?