# mg.metric geometry – Shadows and planar sections of polyhedra – 2

This post continues Shadows and planar sections of polyhedra and On planar sections of 3D convex bodies

Shadows and planar sections of polyhedra gives an example demonstrating that shadows (orthogonal projection of a convex 3D body onto a 2D plane) and planar sections of convex bodies are independent concepts – specifically, the max area shadow of a 3d convex body need not be any planar section at all of the body.

Here, we restate questions asked in On planar sections of 3D convex bodies with “shadow” replacing “planar section” as follows:

Consider the set of shadows of any given convex 3D body.

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

And which convex solid gives it?

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?

Further questions: From among convex bodies for which two quantities x and y from the above set are maximized by the same shadow, will a third quantity z be maximized by another shadow? If so how to quantify this difference? And which convex solid gives it?