Polycentric law is defined as "law arising from a variety of customs and private processes rather than law coercively imposed by a single state authority."
The arguments in favor of such a system are as follows:
1) Morals are physically nonexistent since they are socially constructed. For this reason, moral objectivity in law is not possible because societal norms change over time. Therefore, the law should also change over time, consonant with these norms. Law is thus based on the consensus of the community.
2) Interpretations of law differ across various communities. Thus, no one community can claim to be the sole arbiter of the law, as multiple interpretations exist.
3) Humans cannot be fully objective because everyone's senses are limited in objective perception; for example, humans cannot see certain wavelengths of light and cannot hear certain frequencies of sound. People with schizophrenia perceive a reality that differs from what the consensus perceives. Colorblind people perceive a reality different from non-colorblind people. Objectivity and reason are limited by our limited senses.
One counterargument is that private arbitration groups (clans) unable to resolve their disputes would erupt in gang violence. My response would be that the same thing happens when the government of one country is in dispute with another government; war is almost always the result, except this violence happens on a much larger scale and results in far greater deaths. Read the excerpt below on how polycentric law works well in reality, and protects individual life and property.
One example of such a system that actually works in practice would be Xeer law in Somalia:
Somalia's economy has, in fact, improved as a result of applying Xeer law:
If one starts with premises which preclude the possibility of objective law, one will necessarily attempt to rationalize some alternative -- in this case, a form of law based on social subjectivism.
Objectivism affirms the opposite of the basic premises put forth in this question.
"Morals are physically nonexistent since they are socially constructed."
While the use of the term "physically" above is strange (since morals are principles as opposed to physical things), Objectivism holds that morality exists, fundamentally, to serve each man's physical existence, i.e. his life. To baldly claim that morals are "socially constructed" is to deny that each person must decide for himself what is moral, developing his own moral code based on his own observations. The idea of "social construction" implies that the individual is powerless to develop and adhere to a morality which did not come from his social environment, no matter how deadly that social environment is.
"Interpretations of law differ across various communities. Thus, no one community can claim to be the sole arbiter of the law, as multiple interpretations exist."
This is the legal equivalent of saying "People disagree, therefore there's no one truth." It's an attempt at validating subjectivism. Yes, people disagree, but that doesn't mean that no ideas can be demonstrated to be true, correct, or superior to others. On the whole, a society can progress towards greater truth and understanding, and therefore develop laws which are, to a greater and greater extent, consonant with Man's nature, i.e. appropriate for Man.
Yes, countries progress (and even regress) in different directions, and at different speeds, just as individuals do. It is for this reason that the independence of countries, and of individuals, exists. But such independence requires the enforcement of a principle: the non-initiation of physical force.
The independence of countries is not an affirmation of multi-culturalism. But it is also not true that the affirmation of the objective superiority of a particular legal system implies a right or even an intention to overthrow all other legal systems.
"Humans cannot be fully objective because everyone's senses are limited in objective perception."
This is the most fundamental lie of all. It's a hard one to refute in a short paragraph or two, so I'll simply recommend reading Objectivism: The Philosophy of Ayn Rand, by Leonard Peikoff, specifically chapters one and two. In these chapters he clears away all of the historical confusion surrounding the validity of the senses. Of fundamental importance is understanding the division between perception and conception. To conflate the two is a huge mistake, because it causes one to view conceptual error as something which is built into perception itself.
Conceptual error exists. Perceptual error does not.
There are many more minor errors in the above question, which cannot be addressed here. Again, I must strongly recommend reading Objectivism: The Philosophy of Ayn Rand. There's no other book that reframes philosophy as such, and refutes persistent historical errors better than that book.
I'll caution you, that no book can be expected to make you instantly change your mind. If you strongly believe the premises offered at the top of this question, you'll find yourself disagreeing with Dr. Peikoff. But I hope that in reading his work, you'll sense that someone (you or he) is somehow mistaken, and you'll work at identifying the nature of the error before you simply decide that it is he who is mistaken.