On some variational problems involving capacity, torsional rigidity, perimeter and measure

We investigate the existence of a maximiser among open, bounded, convex sets in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>d</m:mi> </m:msup> </m:math> Rᵈ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> d≥ 3 , for the product of torsional rigidity and Newtonian capacity (or logarithmic capacity if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> d=2 ), with constraints involving Lebesgue measure or a combination of Lebesgue measure and perimeter.