Convex Regions and Quantities - a big bunch of Max and Min questions

References:

1. maximizing and minimizing diameter
2. maximizing and minimizing diameter 2
3. average centralness and depth
4. maximizing and minimizing moment of inertia

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With any convex planar region, we can associate a whole range of quantities:

1. Area
2. Perimeter
3. Diameter
4. Least Width
5. Centralness Coefficient (for definition, see reference 3)
6. Depth Ratio (reference 3)
7. Dimensions of its smallest rectangular bounding box
8. Radius of its circumcircle and largest inscribed circle
9. Dimensions of its least area containing Ellipse and largest inscribed ellipse
10. Moment of inertia about an axis thru the center of mass and perpendicular to the plane of the region

and many more... Indeed, one cannot completely specify a convex region by fixing any finite number of such global measures.

The basic Isoperimetric question (Dido's problem) asks for the convex region of max area among those with specified perimeter. Some more questions have been mentioned and partially answered in the references given above.

The main points one wants to raise in this post:

- There are any number of results (quite a few of them might be unknown as of now) on convex shapes that maximize/minimize any given quantity among those listed above when one or some of the other quantities listed above are fixed.

- And there could he nice new theorems that go: Quantity x is fixed when the list of quantities {....} are fixed.

To achieve maximum area the general trick seems to be to use circular arcs and to increase perimeter, to use straight edges. . Some extreme cases are obvious - eg. if area is specified as pi and perimeter as 2*pi, then, everything else is fixed. What we are asking about are sittuations when these quantities can vary freely and fixing a certain subset of them, nontrivially fixes another quantity.

Smallest n-ellipses that contain convex polygonal regions

This post adds some speculations to https://mathoverflow.net/questions/403370/smallest-3-ellipses-that-contain-triangles.

Are the following claims valid?

"Given any convex polygonal region C, and integer n > m, the smallest n-ellipse that contains C is strictly smaller than the smallest m-ellipse that contains C. And the largest n-ellipse contained entirely in C is strictly larger than the largest m-ellipse contained within C."

Here "larger(smaller)" means larger(smaller) both in terms of area and perimeter.

Remarks: The claims above have exceptions. Indeed, if C is a square, the inscribed circle is both the largest 1-ellipse and largest 2-ellipse contained within C. However, it might well be the case that for larger n, larger n-ellipses contained within the square exist - and if such is not the case, an explanation would be needed. Guess: n-ellipses that are tangent to C at the midpoint of each of C's edges seem good candidates for large inscribed objects.

Refs: https://en.wikipedia.org/wiki/N-ellipse https://web.archive.org/web/20160928200222/http://renyi.mta.hu/~p_erdos/1982-18.pdf