Make a Gradient Follow a path and Fill a Shape

Since this question was posted Adobe have added "Freeform gradient" 

...halfway across the globe. pearl of the orient seas.

This was already answered previously: Making a gradient follow a path

Just to demonstrate step 3 mentioned by tromboniator​

Select the Path,

Activate the stroke color.

Then, Click the 2nd Stroke options as shown below:

Image taken from post : Making a gradient follow a path

MP,

In addition to what Peter said, to create the middle path, you can copy the closed path beneath/on top of itself, then cut the copy path at the sharp ends, then reverse either open path and create a one step blend between them, then expand the blend and delete the outer paths and apply the gradient to the new path.

1. In the Gradient panel build a gradient with the colors you want.

2. With the Pen Tool, draw a path that more or less follows the middle of the crescent shape.

3. Apply the gradient to the stroke of the drawn path, set to go along the path.

4. Set stroke width wide enough to cover widest part of crescent.

5. Send path to back.

6. Select crescent and stroke

7, Menu Object > Clipping Mask > Make