A sharp lower bound for a total curvature functional on closed curves in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">E</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:math> via the Sturm-Hurwitz theorem