A quadrilateral inscribed in a rectangle

Let’s suppose that $X$, $Y$, $Z$, $T$ are in $AB$, $BC$, $CD$, $DA$, respectively. Construct the reflection of $X$ through $AD$ $X_1$, the reflection of $X$ through $BC$ $X_2$, and the reflection of $X_1$ through $CD$ $X_3$. Your diagram should look like this.

Now, we have $$XY+YZ+ZT+TX$$ $$=X_2Y+YZ+ZT+TX_1$$ $$\ge X_2Z+ZX_1$$ $$=X_2Z+ZX_3$$ $$\ge X_3X_2.$$ However, since $X_1X_2=2AB$, $X_1X_3=2AD$, $\angle X_3X_1X_2=\angle DAB=90^\circ$, $$\bigtriangleup X_1X_2X_3\sim\bigtriangleup ABD,$$ so that $$X_3X_2=2AC,$$ and $XY+YZ+ZT+TX\ge 2AC$, as we wanted. $\blacksquare$

Just mirror your rectangle several times.

Take first the special case where $XYZT$ is a parallelogram with sides parallel to diagonals $AC$, $DB$.

Then $XY+ZT=2EF$. And since $\triangle ZCY$ is right and $F$ bisects $ZY$, then $ZY=2FC$. Likewise $TX=2AE$.

Therefore perimeter $P$ of $XYZT=2AC$.

Now let $X'$ be any other point on $AB$. Join $X'T$ and $X'Y$, and through $X$ draw an ellipse with $T$, $Y$ as foci.

Since $\triangle TAX\sim\triangle YBX$, then$$\angle TXA=\angle YXB$$$AB$ is tangent to the ellipse at $X$ (see Apollonius, Conics III, 48), and all other points $X'$ on $AB$ lie outside the ellipse.

And since by the well-known property of an ellipse$$XT+XY=JT+JY=GH$$but$$JX'+X'Y>JY$$therefore$$X'T+X'Y>XT+XY$$Similarly, taking any other point Z' on $CD$, we show that$$Z'T+Z'Y>ZT+ZY$$Therefore, except in the special case first considered, in a quadrilateral inscribed in a rectangle$$P>2AC$$