On Centers of Convex Regions Based On Partitions

These questions are inspired by Yaglom and Boltyanskii's classic, 'Convex Figures'.

Y and B prove this theorem
Winternitz Theorem: If a convex figure is divided into 2 parts by a line l that passes through its center of gravity, the ratio of the areas of the two parts always lies between between the bounds 4/5 and 5/4.

They also prove that for any triangle, there is no other point O than its center of gravity (centroid) for which the ratio of the partial areas into which the triangle is subdivided by lines thru O can be enclosed within narrower bounds.

Question 1: For any general convex 2D region, is the centre of mass still the point such that the areas into which the region is divided by lines thru that points are closest to each other? If the point we seek is not necessarily the centre of mass, then it could be called the 'area partition center' of the region and finding this center for a general given region could be an algorithmic question.

Y and Balso prove:
Let a bounded curve of length L that may consist of separate pieces be given in the plane. Then there is a point O in the plane so that each line through O divides the curve into 2 parts each having a length of not less than L/3.

Question 2: If L be the boundary of a single convex region, there must be a point O' in its interior such that any line thru O' divides the boundary into 2 portions such that the lengths of the two portions are closer than 1:3. What is a bound for this ratio for convex regions?

Let us define the perimeter partition center of a convex region as that point P in its interior such that the 2 portions into which any line thru P divides the outer boundary are guaranteed to be closest to each other in length.

Example: For an isosceles triangle with very narrow base, this perimeter partition center is close to the mid point of the median of its apex and so clearly different from the centroid.

Question 3 Given a general convex region (even a triangle) to find its perimeter partition center.

Note: These questions have 3D analogs with volume and surface area replacing area and perimeter.

An Old Conjecture Disproved - and Another Question

Long ago, I had struggled unsuccessfully with this claim:

For any given number N, if any given convex polygonal region P allows a zero-waste partition into N non-convex congruent pieces each with finitely many sides, then P also allows a partition into N convex congruent pieces with zero wastage.

Eg: https://arxiv.org/pdf/1012.3106.pdf

Now, thanks to Prof. Noam Elkies, we have a very neat counter example. See this discussion:

https://mathoverflow.net/questions/370302/a-claim-on-partitioning-a-convex-planar-region-into-congruent-pieces
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A side excursion into cycloids - an area I know very little about - resulted in this discussion: https://mathoverflow.net/questions/370727/on-cycloids-and-other-roulettes