Higher, greater or bigger distance?

Which of the following is most correct?

The distance of the shortest path must not be higher than 10km.
The distance of the shortest path must not be greater than 10km.
The distance of the shortest path must not be bigger than 10km.

Or is there an even better word?

As height is not being mentioned here, but rather distance on a horizontal scale, "higher" would be inappropriate.

"Bigger" refers to size, not magnitude, and therefore, in this case, is also inappropriate. It's inappropriate because "distance" cannot be measured in size, but in magnitude. "Great length" not "big length".

That leaves, "greater" which is correct.

A better phrasing could be:

The distance of the shortest path must not be more than 10km.

According to this NGram, "greater distance" is the most common:

Next after this is "further", as in "I ran further than he did". You could also say:

The distance of the shortest path must not be further than 10km.

However, I would suggest using "greater" as it seems to fit the context better.

While I agree that "greater than" is correct, I'm not sure about the rest of the sentence. Two places have a distance between them, but does a path have a distance? I don't think it does.

The length of the shortest path must not be greater than 10km

or even

The shortest path must not be longer than 10km

The distance of the shortest path must not be greater than 10km.

This one is correct.

Other workable constructions:

• The length of the shortest path must not be above 10 km.
• The length of the shortest path must not exceed 10 km.

With the latter, you can even omit “the length”, as it’s implied:

• The shortest path must not exceed 10 km.