\eps{\varepsilon}\tldO{O}We study the problem of constructing weak \eps-nets where the stabbing elements are lines or k-flats instead of points. We study this problem in the simplest setting where it is still interesting---namely , the uniform measure of volume over the hypercube [0,1]^d\bigr.. We show that in this setting one can construct (k,\eps) -nets of size m =%DIFDELCMD < \tldO%%% (1/%DIFDELCMD < \eps%%% ^{1-k/d), where }%DIFDELCMD < \tldO %%% hides polylogarithmic factors. That is , there is a set of m k-flats, such that any convex body in [0,1]^d of volume larger than \eps is stabbed by one of these k-flats. \eps\eps). We also prove that any such net must have size at least \Omega(1/}\eps As a concrete example, in three dimensions all \eps-heavy bodies in [0,1]^3 can be stabbed by %DIFDELCMD < \tldO%%%  (1/\eps^{2/3}) lines. Note, that these bounds are sublinear in 1/\eps, and are thus somewhat surprising . We also present deterministic constructions of weak%DIFDELCMD < \eps%%% -nets, under the same volume measure of 0,1%DIFDELCMD < ]%%% ^d. While our bounds are still weaker than known bounds, the new constructions are simple and explicit, unlike previous constructions .
\eps{\varepsilon}\tldO{O}We study the problem of constructing weak \eps-nets where the stabbing elements are lines or k-flats instead of points. We study this problem in the simplest setting where it is still interesting --namely , the uniform measure of volume over the hypercube [0,1]^d\bigr.. Specifically, a (k,\eps) %DIFDELCMD < \tldO%%% %DIFDELCMD < \eps%%% ), where }%DIFDELCMD < \tldO %%% -net is a set of k-flats, such that any convex body in [0,1]^d of volume larger than \eps is stabbed by one of these k-flats. We show that for k \geq 1, one can construct (k,\eps)-nets of size O(1/\eps^{1-k/d). We also prove that any such net must have size at least \Omega(1/}\eps^{1-k/d As a concrete example, in three dimensions all \eps-heavy bodies in [0,1]^3 can be stabbed by %DIFDELCMD < \tldO%%% \Theta (1/\eps^{2/3}) lines. Note, that these bounds are sublinear in 1/\eps, and are thus somewhat surprising %DIFDELCMD < \eps%%% %DIFDELCMD < ]%%% .
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