Let $P$ be a convex polygon (or any convex body in $\mathbb{R}^2$)
with perimeter of length $1$. Call a chord $c$ of $P$ *perimeter-halving*
if half the perimeter lies to one side of $c$
(and so half to the other side).
Here are three convex polygons with many perimeter-halving chords drawn:

(Perimeter-halvings play a role in folding convex polygons to convex polyhedra.)

Define the *perimeter-halving center* for $P$ to be a point $x$
that minimizes the maximum distance $\delta$ of any perimeter-halving chord to $x$.
So the perimeter-halving chords all nearly pass through $x$.

. Does the perimeter-halving center of $P$ coincide with the centroid of $P$? Or is it located at some other natural center?Q1

^{ Center of gravity (cg) marked. $\delta = 0.035$ (from the cg). }

. Which shapes achieve the extremes of $\delta$?Q2

Clearly any centrally symmetric shape achieves $\delta=0$. Does any other shape realize $\delta=0$? Which shapes have the worst (largest) $\delta$?

And just out of curiosity, I would be interested to learn what are the elegant spirograph/astroid-like envelope curves visible in the figures.

**Added**. David Eppstein's center for an equilateral triangle,
as detailed by Wolfgang.

^{ Perimeter-halving center (red); $\delta= \sqrt{3}/72$. }