Smooth Approximation of Lipschitz Functions on Finsler Manifolds
Smooth Approximation of Lipschitz Functions on Finsler Manifolds
We study the smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. We prove that, given a Lipschitz function<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mi>M</mml:mi><mml:mo>→</mml:mo><mml:mi>ℝ</mml:mi></mml:math>defined on a connected, second countable Finsler manifold<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math>, for each positive continuous function<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>ε</mml:mi><mml:mo>:</mml:mo><mml:mi>M</mml:mi><mml:mo>→</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant="normal">∞</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>and each<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>r</mml:mi><mml:mo>></mml:mo><mml:mn …