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Given a set of points in a metric space, the $(k,z)$-clustering problemconsists of finding a set of $k$ points called centers, such that the sum ofdistances raised to the power of $z$ of every data point to its closest centeris minimized. Special cases include the famous k-median problem ($z = 1$) andk-means problem ($z = 2$). The $k$-median and $k$-means problems are at theheart of modern data analysis and massive data applications have given raise tothe notion of coreset: a small (weighted) subset of the input point setpreserving the cost of any solution to the problem up to a multiplicative $(1\pm \varepsilon)$ factor, hence reducing from large to small scale the input tothe problem.  In this paper, we present improved lower bounds for coresets in variousmetric spaces. In finite metrics consisting of $n$ points and doubling metricswith doubling constant $D$, we show that any coreset for $(k,z)$ clusteringmust consist of at least $\Omega(k \varepsilon^{-2} \log n)$ and $\Omega(k\varepsilon^{-2} D)$ points, respectively. Both bounds match previous upperbounds up to polylog factors. In Euclidean spaces, we show that any coreset for$(k,z)$ clustering must consists of at least $\Omega(k\varepsilon^{-2})$points. We complement these lower bounds with a coreset construction consistingof at most $\tilde{O}(k\varepsilon^{-2}\cdot \min(\varepsilon^{-z},k))$ points.

Towards Optimal Lower Bounds for k-median and k-means Coresets