We prove that 3-query linear locally correctable codes over the Reals of dimension $d$ require block length $n>d^{2+\lambda}$ for some fixed, positive $\lambda >0$. Geometrically, this means that if $n$ vectors in $R^d$ are such that each vector is spanned by a linear number of disjoint triples of others, then it must be that $n > d^{2+\lambda}$. This improves the known quadratic lower bounds (e.g. {KdW04, Wood07}). While a modest improvement, we expect that the new techniques introduced in this work will be useful for further progress on lower bounds of locally correctable and decodable codes with more than 2 queries, possibly over other fields as well.

Our proof introduces several new ideas to existing lower bound techniques, several of which work over every field. At a high level, our proof has two parts, {\it clustering} and {\it random restriction}.

The clustering step uses a powerful theorem of Barthe from convex geometry. It can be used (after preprocessing our LCC to be {\it balanced}), to apply a basis change (and rescaling) of the vectors, so that the resulting unit vectors become {\it nearly isotropic}.
This together with the fact that any LCC must have many `correlated' pairs of points, lets us deduce that the vectors must have a surprisingly strong geometric clustering, and hence also combinatorial clustering with respect to the spanning triples.

In the restriction step, we devise a new variant of the dimension reduction technique used in previous lower bounds, which is able to take advantage of the combinatorial clustering structure above. The analysis of our random projection method reduces to a simple (weakly) random graph process, and works over any field.