Geometric Sets of Low Information Content

In this paper we study language classes defined by nonuniform families of hyperplanes and halfspaces. These are subclasses of P/poly. Upto many-one equivalence these classes are essentially the classes ELT1 (and LT1) of languages accepted by nonuniform families of polynomial size depth-1 circuits with a weighted exact threshold gate at the root (respectively, weighted threshold gate at the root). We investigate the consequences of intractable sets being reducible to sets in ELT1 and LT1. Using the polynomial-time algorithm for linear programming and ideas from the theory of convex polytopes we prove that every disjunctively self-reducible bd-cylinder that is many-one reducible to a set in ELT1 or LT1 is already in P. As a consequence we derive that if a complete set for any class K > ϵ {NP, ModkP, PP, C=P} many-one reduces to a set in ELT1 or LT1 then K >= P. For families of hyperplanes over finite fields, we prove that every Turing self-reducible set that is many-one reducible to a set defined by such a family is in P, and every word-decreasing self-reducible set that is many-one reducible to a set defined by such a family is in NP ∩ co-NP. Furthermore, as an interesting connection, it turns out that several sparse and tally reduction classes are many-one reducible to ELT1 or LT1. Thus, our results concerning reductions to ELT1 and LT1 subsume the strongest existing collapse results concerning reductions of any class K > ϵ {NP, ModkP, PP, C=P} to sparse sets for various reducibilities.