jats:titleAbstract</jats:title>jats:pFor jats:inline-formulajats:alternativesjats:tex-math$$\gamma \in (0,2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo∈</mml:mo> mml:mo(</mml:mo> mml:mn0</mml:mn> mml:mo,</mml:mo> mml:mn2</mml:mn> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, we define a jats:italicweak</jats:italic>jats:inline-formulajats:alternativesjats:tex-math$$\gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miγ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>-jats:italicLiouville quantum gravity</jats:italic> (jats:italicLQG</jats:italic>) jats:italicmetric</jats:italic> to be a function jats:inline-formulajats:alternativesjats:tex-math$$h\mapsto D_h$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mih</mml:mi> mml:mo↦</mml:mo> mml:msub mml:miD</mml:mi> mml:mih</mml:mi> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> which takes in an instance of the planar Gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms. We show that these axioms are satisfied for any subsequential limits of Liouville first passage percolation. Such subsequential limits were proven to exist by Ding et al. (Tightness of Liouville first passage percolation for jats:inline-formulajats:alternativesjats:tex-math$$\gamma \in (0,2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo∈</mml:mo> mml:mo(</mml:mo> mml:mn0</mml:mn> mml:mo,</mml:mo> mml:mn2</mml:mn> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, 2019. ArXiv e-prints, <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" ext-link-type="uri" xlink:href="http://arxiv.org/abs/1904.08021">arXiv:1904.08021</jats:ext-link>). It is also known that these axioms are satisfied for the jats:inline-formulajats:alternativesjats:tex-math$$\sqrt{8/3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msqrt mml:mrow mml:mn8</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula>-LQG metric constructed by Miller and Sheffield (2013–2016). For any weak jats:inline-formulajats:alternativesjats:tex-math$$\gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miγ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>-LQG metric, we obtain moment bounds for diameters of sets as well as point-to-point, set-to-set, and point-to-set distances. We also show that any such metric is locally bi-Hölder continuous with respect to the Euclidean metric and compute the optimal Hölder exponents in both directions. Finally, we show that LQG geodesics cannot spend a long time near a straight line or the boundary of a metric ball. These results are used in subsequent work by Gwynne and Miller which proves that the weak jats:inline-formulajats:alternativesjats:tex-math$$\gamma $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miγ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>-LQG metric is unique for each jats:inline-formulajats:alternativesjats:tex-math$$\gamma \in (0,2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo∈</mml:mo> mml:mo(</mml:mo> mml:mn0</mml:mn> mml:mo,</mml:mo> mml:mn2</mml:mn> mml:mo)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, which in turn gives the uniqueness of the subsequential limit of Liouville first passage percolation. However, most of our results are new even in the special case when jats:inline-formulajats:alternativesjats:tex-math$$\gamma =\sqrt{8/3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miγ</mml:mi> mml:mo=</mml:mo> mml:msqrt mml:mrow mml:mn8</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>