Besicovitch’s \frac{1}2 problem and linear programming

We consider the following classical conjecture of Besicovitch: a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional Borel set in the plane with finite Hausdorff <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {H}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which has lower density strictly larger than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="one half"> <mml:semantics> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {1}{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> almost everywhere must be countably rectifiable. We improve the best known bound, due to Schechter, showing that the statement is indeed true if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="one half"> <mml:semantics> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {1}{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is replaced by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="seven tenths"> <mml:semantics> <mml:mfrac> <mml:mn>7</mml:mn> <mml:mn>10</mml:mn> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {7}{10}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Concerning Schechter’s bound, which is in fact an improvement of an argument by Preiss and Tišer, and it is valid in any metric space, we give a simpler proof. More importantly, we propose a family of variational problems to produce the latter and many other similar bounds and we study several properties of them, paving the way for further improvements.