# Plotting regions with 1D and 2D components

I am trying to plot a region defined by an inequality which contains both a 2D square and a 1D segment. My issue is that RegionPlot only shows the square and ignores the line.

I noticed that if I use Reduce on the inequality, the solution is complete and contains the 1D region, and then I can use ImplicitRegion and DiscretizeRegion to visualize it correctly, but my goal is to represent the region in the xy plane with some interactive features.

This is the function I use in the inequality:

dv[{p1_, p2_}, {x_, y_}] := Piecewise[{
{Abs[p2 - y], p1 == x},
{Abs[p2] + Abs[x - p1] + Abs[y], p1 != x}
}]

This is the interactive RegionPlot which doesn't show the 1D line:

Manipulate[
DynamicModule[{center = {0, 0}},
LocatorPane[Dynamic[center], Dynamic@RegionPlot[
dv[center, {x, y}] <= \[Epsilon],
{x, -10, 10}, {y, -10, 10},
Sequence[
Axes -> True, AxesStyle -> Thick, GridLines -> Automatic,
BoundaryStyle -> Directive[
Thickness[Medium], Dashed], MaxRecursion -> 3,
PerformanceGoal -> "Quality"]
, ImageSize -> Large]]],
{\[Epsilon], 0, 10}
]

And this is the correct solution given by Reduce, along with its dicretized representation:

sol = Reduce[dv[{1, 3}, {x, y}] < 6, Reals]

(-2 < x < 1 && -2 - x < y < 2 + x) || (x == 1 && -3 < y < 9) || (1 <
x < 4 && -4 + x < y < 4 - x)

region = ImplicitRegion[sol, {x, y}];
DiscretizeRegion[region]

I am not an expert in using Mathematica so probably there is an obvious way to fix this. Thanks in advance.

Clear["Global`*"]

dv[{p1_, p2_}, {x_, y_}] :=
Piecewise[{{Abs[p2 - y], p1 == x}, {Abs[p2] + Abs[x - p1] + Abs[y],
p1 != x}}]

Manipulate[
Module[{sol, region},
center = Rationalize@center;
sol = Reduce[dv[center, {x, y}] < Rationalize[\[Epsilon]], Reals];
region = ImplicitRegion[sol, {x, y}];
DiscretizeRegion[region,
PlotRange -> 15,
Axes -> False,
Frame -> True,
GridLines -> Automatic,
GridLinesStyle -> Red,
PerformanceGoal -> "Quality",
ImageSize -> Medium]],
{{\[Epsilon], 5}, 0, 10, 0.1, Appearance -> "Labeled"},
{{center, {2, 2}}, Locator}]