$L_p$ normalization is actually described in Wikipedia that quotes Dalal and Triggs (2005)
Dalal and Triggs explored four different methods for block normalization. Let $v$ be the non-normalized vector containing all histograms in a given block, $\|v\|_k$ be its $k$-norm for $k={1,2}$ and $e$ be some small constant (the exact value, hopefully, is unimportant). Then the normalization factor can be one of the following:
L2-norm: $f = {v \over \sqrt{\|v\|^2_2+e^2}}$
(...)
L1-norm: $f = {v \over (\|v\|_1+e)}$
L1-sqrt: $f = \sqrt{v \over (\|v\|_1+e)}$
So $L_p$ normalization of histograms in fact relates to using $L_p$ norms$L_p$ norms to normalize vectors.
By "dimensions" authors mean the "length" of vectors.
Dalal, N., & Triggs, B. (2005). Histograms of oriented gradients for human detection. In 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) (Vol. 1, pp. 886-893). IEEE.