Does the positive constant $c_d$, depending only from dimension, with the following property exist?

Property: for every convex body $K\subset \mathbb R^d$ exists parallelepiped $P\subset K$ so that $$ |P\cap \mathbb Z^d| \ge c_d |K\cap \mathbb Z^d|. $$

Does the positive constant $c_d$, depending only from dimension, with the following property exist?

Property: for every convex body $K\subset \mathbb R^d$ there exists parallelepiped $P\subset K$ so that $$ |P\cap \mathbb Z^d| \ge c_d |K\cap \mathbb Z^d|. $$

Does exist positive constant $c_d$, depending only from dimension with the property: for every convex body $K\subset \mathbb R^d$ exists parallelepiped $P\subset K$ so that $$ |P\cap \mathbb Z^d| \ge c_d |K\cap \mathbb Z^d|? $$

Does the positive constant $c_d$, depending only from dimension, with the following property exist?

Property: for every convex body $K\subset \mathbb R^d$ exists parallelepiped $P\subset K$ so that $$ |P\cap \mathbb Z^d| \ge c_d |K\cap \mathbb Z^d|. $$