Classical clustering methods usually face tough challenges when we have a larger set of features compared to the number of items to be partitioned. We propose a Sparse MinMax k-Means Clustering approach by reformulating the objective of the MinMax k-Means algorithm (a variation of classical k-Means that minimizes the maximum intra-cluster variance instead of the sum of intra-cluster variances), into a new weighted between-cluster sum of squares (BCSS) form. We impose sparse regularization on these weights to make it suitable for high-dimensional clustering. We seek to use the advantages of the MinMax k-Means algorithm in the high-dimensional space to generate good quality clusters. The efficacy of the proposal is showcased through comparison against a few representative clustering methods over several real world datasets.