On the sum of squares of cell complexities in hyperplane arrangements

Let H be a collection of n hyperplanes in IRd, d ~ 2. For each cell c of the arrangement of H let ~(c) denote the number of faces of c of all dimensions. We prove that XC ~(c)z = O(nd log[~j -1 n), where the sum extends over all cells of the arrangement. Among other applications, we show that the total number of faces bounding any m distinct cells in an arran ement of f n hyperplanes in IRd is 0(rnl/2nd/2 log( [d 2J‘lJf2 n) and provide a lower bound on the maximum possible face count in m distinct cells, which is close to the upper bound, and for many values of m and n is fl(m1J2nd12). ● Work on this paper by the third author has been supported by Office of Naval Research Grant NOO014-90-.J-12S4, by National Science Foundation Grant CCR-89-01484, and by grants from the U.S.-Israeli Binationaf Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G .I.F., the German-Israeli Foundation for Scientific Research and Development. tDepartment of Computer Science, Polytechnic University, Brooklyn, NY 11201 USA $Department of Applied Mathematics, Charles University, 11800 Praha 1, Czechoslovakia sS&CIOl of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel and Courant Institute of Mathematical Sciences, New York University, NY 10012 USA Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notiec and the title of the publication and its date appear, and notice is given that copying N by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific pennission~” o 1991 ACM 0.89791 -426-0/91/0006/0307 $1.50