documentclass{amsart}
usepackage{amsmath}
usepackage{amsfonts}

usepackage{tikz}
usetikzlibrary{calc,angles,positioning,intersections}


begin{document}


noindent hspace*{fill}
begin{tikzpicture}[node distance= 3mm and 0mm]

%A hexagon - a set that is not convex - is drawn which is symmetric across the x-axis.
coordinate (A) at (0,0);
coordinate (O) at (-5,0);

path[name path=x-axis] (A) -- (O);
path[name path=horizontal_line_at_-1] (-5,-1) -- (0,-1);
path[name path=horizontal_line_at_1] (-5,1) -- (0,1);

coordinate (B) at ($(A) + (-135:2.5)$);
draw (A) -- (B);

coordinate (C') at ($(B) + (165:3)$);
path[name path=extension_of_line_segment_BC] (B) -- (C');
coordinate[name intersections={of=extension_of_line_segment_BC and horizontal_line_at_-1, by={C}}];
draw (B) -- (C);

coordinate (D') at ($(C) + (45:2)$);
path[name path=extension_of_line_segment_CD] (C) -- (D');
coordinate[name intersections={of=extension_of_line_segment_CD and x-axis, by={D}}];
draw (C) -- (D);

coordinate (E') at ($(D) + (135:2)$);
path[name path=extension_of_line_segment_DE] (D) -- (E');
coordinate[name intersections={of=extension_of_line_segment_DE and horizontal_line_at_1, by={E}}];
draw (D) -- (E);

coordinate (F) at ($(E) + (15:3)$);
draw (E) -- (F);
draw (A) -- (F);


%Points P and Q in the hexagon are plotted. Line segment $overline{PQ}$ is not contained in the
%hexagon.
coordinate (P) at (-4,0.75);
draw[fill] (P) circle (1.5pt);
coordinate (Q) at (-4,-0.75);
draw[fill] (Q) circle (1.5pt);
draw (P) -- (Q);
%Points P and Q are labeled.
coordinate (label_for_P) at ($(P)!-3mm!-90:(Q)$);
node at (label_for_P){$P$};
coordinate (label_for_Q) at ($(Q)!-3mm!90:(P)$);
node at (label_for_Q){$Q$};

%Title for hexagon is typeset.
node[font=bfseries,align=center,above=of F -| {$(C |- A)!0.5!(A)$}] (title) {A set that is \ not convex};


%A convex pentagon is drawn which is symmetric across the x-axis.
begin{scope}[transform canvas={xshift=6cm}]
coordinate (A) at (0,0);
coordinate (O) at (-5,0);

path[name path=x-axis] (A) -- (O);
path[name path=horizontal_line_at_-1] (-5,-1) -- (0,-1);
path[name path=horizontal_line_at_1] (-5,1) -- (0,1);

coordinate (B) at ($(A) + (-135:2.5)$);
draw (A) -- (B);

coordinate (C') at ($(B) + (165:3)$);
path[name path=extension_of_line_segment_BC] (B) -- (C');
coordinate[name intersections={of=extension_of_line_segment_BC and horizontal_line_at_-1, by={C}}];
draw (B) -- (C);

coordinate (D') at ($(C) + (45:2)$);
path[name path=extension_of_line_segment_CD] (C) -- (D');
coordinate[name intersections={of=extension_of_line_segment_CD and x-axis, by={D}}];
draw[dashed] (C) -- (D);

coordinate (E') at ($(D) + (135:2)$);
path[name path=extension_of_line_segment_DE] (D) -- (E');
coordinate[name intersections={of=extension_of_line_segment_DE and horizontal_line_at_1, by={E}}];
draw[dashed] (D) -- (E);

draw (C) -- (E);

coordinate (F) at ($(E) + (15:3)$);
draw (E) -- (F);
draw (A) -- (F);

%Title for pentagon is typeset.
node[font=bfseries, above=of E -| {$(C |- A)!0.5!(A)$}] (title) {A convex set};
end{scope}
end{tikzpicture}
hspace{fill}

end{document}